(τ₁,τ₂)δ-semicontinuous multifunctions

Abstract

The purpose of the present article is to introduce the concepts of upper and lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunctions. Some characterizations of upper and lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunctions are investigated. The relationships between upper and lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunctions and the other types of continuity for multifunctions are established.

Bitopological space; Semi-continuity; Almost continuity; Weak continuity; Multifunction

1. Introduction

Continuity is an important concept for the study and investigation in topological spaces. This concept has been extended to the setting of multifunctions and has been generalized by weaker forms of open sets. In 1963, Levine [16] introduced the notions of semi-open sets and semi-continuity in topological spaces. It is shown in [19] that semi-continuity is equivalent to quasicontinuity due to Marcus [17]. Veličko [32] introduced δ-open sets, which are stronger than open sets, in order to investigate the characterization of H-closed spaces. Park et al. [24] have introduced the notion of δ-semiopen sets which are stronger than semi-open sets but weaker than δ-open sets and investigated the relationships between several types of these open sets. Caldas et al. [4] and [3] investigated this class of sets further and also studied some of its applications. In [15], the present authors obtained the further properties of δ-semiopen sets and related sets. Ekici [9] introduced the notions of δ-semi-generalized closed sets, locally δ-semi-generalized closed sets, $\text{l}\delta \text{sg}{\text{c}}^{⁎}$-sets and $\text{l}\delta \text{sg}{\text{c}}^{⁎⁎}$-sets. The class of δ-semi-generalized closed sets contain the classes of δg-closed sets and δ-semiclosed sets. The notions of locally δ-semi-generalized closed sets, $\text{l}\delta \text{sg}{\text{c}}^{⁎}$-sets and $\text{l}\delta \text{sg}{\text{c}}^{⁎⁎}$-sets are weaker forms of locally δ-generalized closed sets, $\text{l}\delta \text{g}{\text{c}}^{⁎}$-sets and $\text{l}\delta \text{g}{\text{c}}^{⁎⁎}$-sets, respectively.

In 1980, Noiri [22] introduced a new class of functions called δ-continuous and investigated the relationships between δ-continuity and near-compactness due to Singal and Asha Mathur [28]. The concepts of continuity and δ-continuity are independent of each other and both imply almost-continuity due to Singal and Singal [29]. However, an almost-continuous function need not be δ-continuous. Ekici [12] introduced and investigated the notion of almost δ-semicontinuous functions which generalize R-maps and δ-continuous functions. Furthermore, Ekici and Noiri [7] investigated the concept of almost δ-semicontinuous functions and proved that a function $f:(X,\tau )\to (Y,\sigma )$ is almost δ-semicontinuous if and only if $f:(X,{\tau }_s)\to (Y,{\sigma }_s)$ is semi-continuous, where ${\tau }_s$ and ${\sigma }_s$ are the semiregularizations of τ and σ, respectively. In 2008, Ekici and Jafari [8] introduced the concept of completely δ-semi-irresolute functions which is weaker form of complete irresolute functions and investigated some of the properties of completely δ-semi-irresolute functions. Ekici [13] defined two new classes of contra-continuity called contra R-continuity and (δ-semi,s)-continuity. (δ-semi,s)-continuity is strictly between contra R-continuity and weakly θ-irresoluteness. Moreover, (δ-semi,s)-continuity generalize perfectly continuity, s-continuity, almost s-continuity and contra R-continuity.

The concepts of preopen sets and precontinuity in topological spaces were first introduced and investigated by Mashhour et al. [18]. Precontinuity was also called almost-continuity in the sense of Husain [14]. Przemski [27] and the present authors [21] have independently defined the notion of precontinuity in the setting of multifunctions. In [25], the authors have shown that these notions are equivalent of each other and obtained several characterizations of precontinuous multifunctions. On the other hand, in [20], the present authors have introduced the notion of weakly precontinuous multifunctions. Park et al. [23] introduced and studied δ-precontinuous multifunctions as a generalization of precontinuous multifunctions due to Popa [26]. Ekici [11] introduced the notion of almost δ-precontinuous multifunctions and investigated several characterizations of almost δ-precontinuous multifunctions. Moreover, Ekici [10] introduced a new form of continuous multifunctions, called upper (lower) δ-semicontinuous multifunctions and obtained some characterizations and relationships among the other related multifunctions. In [6], the present author introduced the notion of almost δ-semicontinuous multifunctions and investigated the relationships among δ-semicontinuity, almost δ-semicontinuity and weak δ-semicontinuity for multifunctions. Recently, Carpintero et al. [5] introduced a new class of multifunctions namely $(\delta ,\omega )$-continuous multifunctions and obtained some characterizations of such multifunctions.

The article is organized as follows. In Section 3, we introduce the notions of upper and lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunctions and investigate some characterizations of such multifunctions. Section 4 is devoted to introducing and studying upper and lower almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunctions. In Section 5, several interesting characterizations of upper and lower weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunctions are investigated. Furthermore, the relationships between $({\tau }_1,{\tau }_2)\delta$-semicontinuity, almost $({\tau }_1,{\tau }_2)\delta$-semicontinuity and weak $({\tau }_1,{\tau }_2)\delta$-semicontinuity are discussed.

2. Preliminaries

Throughout the present paper, spaces $(X,{\tau }_1,{\tau }_2)$ and $(Y,{\sigma }_1,{\sigma }_2)$ (or simply X and Y) always mean bitopological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a bitopological space $(X,{\tau }_1,{\tau }_2)$. The closure of A and the interior of A with respect to ${\tau }_i$ is denoted by ${\tau }_i\text{-Cl}(A)$ and ${\tau }_i\text{-Int}(A)$, respectively, for $i=1,2$. A subset A of a bitopolgical space $(X,{\tau }_1,{\tau }_2)$ is said to be ${\mathit{\tau }}_1{\mathit{\tau }}_2$-closed [2] if $A={\tau }_1\text{-Cl}({\tau }_2\text{-Cl}(A))$. The complement of a ${\tau }_1{\tau }_2$-closed is said to be ${\mathit{\tau }}_1{\mathit{\tau }}_2$-open. The intersection of all ${\tau }_1{\tau }_2$-closed sets containing A is called the ${\mathit{\tau }}_1{\mathit{\tau }}_2$-closure of A and denoted by ${\tau }_1{\tau }_2\text{-Cl}(A)$. The union of all ${\tau }_1{\tau }_2$-open sets contained in A is called the ${\mathit{\tau }}_1{\mathit{\tau }}_2$-interior of A and denoted by ${\tau }_1{\tau }_2\text{-Int}(A)$. A subset N of a bitopological space $(X,{\tau }_1,{\tau }_2)$ is said to be a ${\mathit{\tau }}_1{\mathit{\tau }}_2$-neighborhood of $x\in X$ if there exists a ${\tau }_1{\tau }_2$-open set V of X such that $x\in V\subseteq N$.

A subset A of a topological space $(X,\tau )$ is said to be regular open if $A=\text{Int}(\text{Cl}(A))$ [31]. The complement of a regular open set is said to be regular closed. A subset A is said to be δ-open [32] if for each $x\in A$, there exists a regular open set G such that $x\in G\subseteq A$. A point $x\in X$ is called a δ-cluster point of A if $\text{Int}(\text{Cl}(V))\cap A\ne \varnothing$ for every open set V containing x. The set of all δ-cluster points of A is called the δ-closure of A and is denoted by ${\text{Cl}}_{\delta }(A)$. A subset A of X is called δ-closed if $A={\text{Cl}}_{\delta }(A)$. The set

$$\{x\in X|x\in U\subseteq A\text{for some regular open set}U\text{of}X\}$$

is called the δ-interior of A and is denoted by ${\text{Int}}_{\delta }(A)$.

A subset A of a topological space $(X,\tau )$ is said to be δ-semiopen [24] if there exists a δ-open set U of X such that $U\subseteq A\subseteq \text{Cl}(U)$. The complement of a δ-semiopen set is called δ-semiclosed. A point $x\in X$ is called the δ-semicluster point of A if $A\cap U\ne \varnothing$ for every δ-semiopen set U of X containing x. The set of all δ-semicluster points of A is called the δ-semiclosure of A and is denoted by ${\text{sCl}}_{\delta }(A)$.

By a multifunction $F:X\to Y$, we mean a point-to-set correspondence from X into Y, and we always assume that $F(x)\ne \varnothing$ for all $x\in X$. For a multifunction $F:X\to Y$, following [1], we shall denote the upper and lower inverse of a set B of Y by $F^{+}(B)$ and $F^{-}(B)$, respectively, that is, $F^{+}(B)=\{x\in X|F(x)\subseteq B\}$ and $F^{-}(B)=\{x\in X|F(x)\cap B\ne \varnothing \}$. In particular, $F^{-}(y)=\{x\in X|y\in F(x)\}$ for each point $y\in Y$. For each $A\subseteq X$, $F(A)={\cup }_{x\in A}F(x)$. Then F is said to be surjection if $F(X)=Y$, or equivalent, if for each $y\in Y$ there exists $x\in X$ such that $y\in F(x)$ and F is called injection if $x\ne y$ implies $F(x)\cap F(y)=\varnothing$.

Definition 2.1

A subset A of a bitopological space $(X,{\tau }_1,{\tau }_2)$ is said to be $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semiopen if $A\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(A))$. The complement of a $({\tau }_1,{\tau }_2)\delta$-semiopen set is said to be $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semiclosed.

Definition 2.2

Let A be a subset of a bitopological space $(X,{\tau }_1,{\tau }_2)$. A point $x\in X$ is called the $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicluster point of A if $A\cap U\ne \varnothing$ for every $({\tau }_1,{\tau }_2)\delta$-semiopen set U containing x. The set of all $({\tau }_1,{\tau }_2)\delta$-cluster points of A is called the $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-cluster of A and is denoted by $({\tau }_1,{\tau }_2)\delta \text{-sCl}(A)$. The union of all $({\tau }_1,{\tau }_2)\delta$-semiopen sets contained in A is called the $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semiinterior of A and is denoted by $({\tau }_1,{\tau }_2)\delta \text{-sInt}(A)$.

Lemma 2.3

Let $(X,{\tau }_1,{\tau }_2)$ be a bitopological space and $\{A_{\gamma }|\gamma \in \mathrm{\Gamma }\}$ be a family of subsets of X.

• (1)If $A_{\gamma }$ is $({\tau }_1,{\tau }_2)\delta$-semiopen for each $\gamma \in \mathrm{\Gamma }$, then $\underset{\gamma \in \mathrm{\Gamma }}{\cup }{A}_{\gamma }$ is $({\tau }_1,{\tau }_2)\delta$-semiopen.
• (2)If $A_{\gamma }$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed for each $\gamma \in \mathrm{\Gamma }$, then $\underset{\gamma \in \mathrm{\Gamma }}{\cap }{A}_{\gamma }$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed.

Corollary 2.4

For a subset A of a bitopological space $(X,{\tau }_1,{\tau }_2)$, the following properties hold:

• (1)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(A)=\cap \{F|F\mathit{\text{is}}({\tau }_1,{\tau }_2)\delta \mathit{\text{-semiclosed}}\mathit{\text{and}}A\subseteq F\}$;
• (2)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(A)$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed;
• (3)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(A))=({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(A)$;
• (4)A is $({\tau }_1,{\tau }_2)\delta$-semiclosed if and only if $A=({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(A)$.

Lemma 2.5

For a subset A of a bitopological space $(X,{\tau }_1,{\tau }_2)$, the following properties hold:

• (1)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(A)$ is $({\tau }_1,{\tau }_2)\delta$-semiopen;
• (2)A is $({\tau }_1,{\tau }_2)\delta$-semiopen if and only if $A=({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(A)$;
• (3)$X-({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(A)=({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(X-A)$;
• (4)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(X-A)=X-({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(A)$.

3. On characterizations of upper and lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunctions

In this section, we introduce the notions of upper and lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunctions. Moreover, several interesting characterizations of upper and lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunctions are discussed.

Definition 3.1

A multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is said to be:

• (1)upper $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicontinuous at a point $x\in X$ if for each ${\sigma }_1{\sigma }_2$-open subset V of Y such that $F(x)\subseteq V$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq V$;
• (2)lower $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicontinuous at a point $x\in X$ if for each ${\sigma }_1{\sigma }_2$-open subset V of Y such that $F(x)\cap V\ne \varnothing$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(z)\cap V\ne \varnothing$ for every $z\in U$;
• (3)upper (resp. lower) $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicontinuous if F has this property at each point of X.

Lemma 3.2

For a subset A of a bitopological space $(X,{\tau }_1,{\tau }_2)$, the following properties are hold:

• (1)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(A)={\tau }_1\mathit{\text{-Int}}({\tau }_2{\mathit{\text{-Cl}}}_{\delta }(A))\cup A$.
• (2)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(A)={\tau }_1\mathit{\text{-Cl}}({\tau }_2{\mathit{\text{-Int}}}_{\delta }(A))\cap A$.

Proof

(1) Since $({\tau }_1,{\tau }_2)\delta \text{-sCl}(A)$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed, we have

$${\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(({\tau }_1,{\tau }_2)\delta \text{-sCl}(A)))\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sCl}(A)$$

and hence ${\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(A))\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sCl}(A)$. Thus,

$${\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(A))\cup A\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sCl}(A).$$

To establish the opposite inclusion, we observe that

$${\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }({\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(A))\cup A))\subseteq {\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }({\tau }_2{\text{-Cl}}_{\delta }(A)\cup A))={\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(A))\subseteq {\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(A))\cup A$$

and hence ${\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(A))\cup A$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed. Thus,

$$({\tau }_1,{\tau }_2)\delta \text{-}\text{sCl}(A)\subseteq {\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(A))\cup A.$$

Consequently, we obtain $({\tau }_1,{\tau }_2)\delta \text{-sCl}(A)={\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(A))\cup A$.

(2) By (1), we have

$$({\tau }_1,{\tau }_2)\delta \text{-sInt}(A)=X-({\tau }_1,{\tau }_2)\delta \text{-sCl}(X-A)=X-[{\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(X-A))\cup (X-A)]=X-[(X-{\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(A)))\cup (X-A)]=X-[X-({\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(A))\cap A)]={\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(A))\cap A.$$

 □

Theorem 3.3

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous at $x\in X$;
• (2)$x\in ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{+}(V))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y containing $F(x)$;
• (3)$x\in {\tau }_1\mathit{\text{-Cl}}({\tau }_2{\mathit{\text{-Int}}}_{\delta }(F^{+}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y containing $F(x)$.

Proof

$(1)\Rightarrow (2)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $F(x)$. Since F is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous at x, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq V$. Thus, $x\in U\subseteq F^{+}(V)$. Since U is $({\tau }_1,{\tau }_2)\delta$-semiopen, we have $x\in U=({\tau }_1,{\tau }_2)\delta \text{-sInt}(U)\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(V))$.

$(2)\Rightarrow (3)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $F(x)$. By (2) and Lemma 3.2(2),

$$x\in ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(V))={\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(F^{+}(V)))\cap F^{+}(V)\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(F^{+}(V))).$$

$(3)\Rightarrow (1)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $F(x)$. Then $x\in F^{+}(V)$ and by (3), we have $x\in {\tau }_1\text{-Cl}({\tau }_2\text{-Int}\delta (F^{+}(V)))$. By Lemma 3.2(2), $x\in ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(V))$. Put $U=({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(V))$. Then U is a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq V$. This shows that F is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous at x. □

Theorem 3.4

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous at $x\in X$;
• (2)$x\in ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{-}(V))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y such that $F(x)\cap V\ne \varnothing$;
• (3)$x\in {\tau }_1\mathit{\text{-Cl}}({\tau }_2{\mathit{\text{-Int}}}_{\delta }(F^{-}(V))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y such that $F(x)\cap V\ne \varnothing$.

Proof

The proof is similar to that of Theorem 3.3. □

Definition 3.5

A function $f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is said to be $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicontinuous at a point $x\in X$ if for each ${\sigma }_1{\sigma }_2$-open subset V of Y such containing $f(x)$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $f(U)\subseteq V$. If f has this property at each point of X, then f is said to be $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicontinuous.

Corollary 3.6

For a function $f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)f is $({\tau }_1,{\tau }_2)\delta$-semicontinuous at $x\in X$;
• (2)$x\in ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(f^{-1}(V))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y containing $f(x)$;
• (3)$x\in {\tau }_1\mathit{\text{-Cl}}({\tau }_2{\mathit{\text{-Int}}}_{\delta }(f^{-1}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y containing $f(x)$.

Definition 3.7

A subset B of a bitopological space $(X,{\tau }_1,{\tau }_2)$ is said to be a $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semineighborhood of $x\in X$ if there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset V of X such that $x\in V\subseteq B$.

Theorem 3.8

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)$F^{+}(V)$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (3)$F^{-}(K)$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (4)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}(B))\subseteq F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(B))$ for every subset B of Y;
• (5)for each $x\in X$ and each ${\sigma }_1{\sigma }_2$-neighborhood V of $F(x)$, $F^{+}(V)$ is a $({\tau }_1,{\tau }_2)\delta$-semineighborhood of x;
• (6)for each $x\in X$ and each ${\tau }_1{\tau }_2$-neighborhood V of $F(x)$, there exists a $({\tau }_1,{\tau }_2)\delta$-semineighborhood U of x such that $F(U)\subseteq V$;
• (7)$F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(B))\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{+}(B))$ for every subset B of Y;
• (8)$F^{+}(V)\subseteq {\tau }_1\mathit{\text{-Cl}}({\tau }_2{\mathit{\text{-Int}}}_{\delta }(F^{+}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y.

Proof

$(1)\Rightarrow (2)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y and $x\in F^{+}(V)$. Since F is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq V$. Since $x\in U\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{Int}}_{\delta }(U))\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{Int}}_{\delta }(F^{+}(V)))$, we have $F^{+}(V)\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{Int}}_{\delta }(F^{+}(V)))$ and hence $F^{+}(V)$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X.

$(2)\iff (3)$: It follows from the fact that $F^{+}(Y-B)=X-F^{-}(B)$ for every subset B of Y.

$(3)\Rightarrow (4)$: Let B be any subset of Y. Then ${\sigma }_1{\sigma }_2\text{-Cl}(B)$ is ${\sigma }_1{\sigma }_2$-closed in Y and by (3), $F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(B))$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X. Thus,

$$({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}(B))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(B)).$$

$(4)\Rightarrow (3)$: Let K be any ${\sigma }_1{\sigma }_2$-closed subset of Y. By (4), $({\tau }_1,{\tau }_2)\delta \text{-}\text{sCl}(F^{-}(K))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(K))=F^{-}(K)$ and hence $F^{-}(K)$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X.

$(2)\Rightarrow (5)$: Let $x\in X$ and V be a ${\sigma }_1{\sigma }_2$-neighborhood of $F(x)$. There exists a ${\sigma }_1{\sigma }_2$-open subset G of Y such that $F(x)\subseteq G\subseteq V$. Then $x\in F^{+}(G)\subseteq F^{+}(V)$. By (2), we have $F^{+}(G)$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X and hence $F^{+}(V)$ is a $({\tau }_1,{\tau }_2)\delta$-semineighborhood of x.

$(5)\Rightarrow (6)$: Let $x\in X$ and V be a ${\sigma }_1{\sigma }_2$-neighborhood of $F(x)$. Put $U=F^{+}(V)$. By (5), we have U is a $({\tau }_1,{\tau }_2)\delta$-semineighborhood of x and $F(U)\subseteq V$.

$(6)\Rightarrow (1)$: Let $x\in X$ and V be any ${\sigma }_1{\sigma }_2$-open subset of Y such that $F(x)\subseteq V$. Then V is a ${\sigma }_1{\sigma }_2$-neighborhood of $F(x)$ and by (6), there exists a $({\tau }_1,{\tau }_2)\delta$-semineighborhood U of x such that $F(U)\subseteq V$. Therefore, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset W of X such that $x\in W\subseteq U$ and hence $F(W)\subseteq V$. This shows that F is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

$(2)\Rightarrow (7)$: Let B be any subset of Y. Then ${\sigma }_1{\sigma }_2\text{-Int}(B)$ is ${\sigma }_1{\sigma }_2$-open in Y. By (2), we have $F^{+}({\sigma }_1{\sigma }_2\text{-Int}(B))$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X and hence

$$F^{+}({\sigma }_1{\sigma }_2\text{-Int}(B))\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(B)).$$

$(7)\Rightarrow (2)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y. By (7),

$$F^{+}(V)=F^{+}({\sigma }_1{\sigma }_2\text{-Int}(V))\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(V))$$

and hence $F^{+}(V)$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X.

$(2)\iff (8)$: It follows immediately from definition. □

Theorem 3.9

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)$F^{-}(V)$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (3)$F^{+}(K)$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (4)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}(B))\subseteq F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(B))$ for every subset B of Y;
• (5)for each $x\in X$ and each ${\sigma }_1{\sigma }_2$-neighborhood V which intersects $F(x)$, $F^{-}(V)$ is a $({\tau }_1,{\tau }_2)\delta$-semineighborhood of x;
• (6)for each $x\in X$ and each ${\tau }_1{\tau }_2$-neighborhood V which intersects $F(x)$, there exists a $({\tau }_1,{\tau }_2)\delta$-semineighborhood U of x such that $F(z)\cap V\ne \varnothing$ for each $z\in U$;
• (7)$F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(B))\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{-}(B))$ for every subset B of Y;
• (8)$F^{-}(V)\subseteq {\tau }_1\mathit{\text{-Cl}}({\tau }_2{\mathit{\text{-Int}}}_{\delta }(F^{-}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y.

Proof

The proof is similar to that of Theorem 3.8. □

Corollary 3.10

For a function $f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)f is $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)$f^{-1}(V)$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (3)$f^{-1}(K)$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (4)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}(B))\subseteq f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(B))$ for every subset B of Y;
• (5)for each $x\in X$ and each ${\sigma }_1{\sigma }_2$-neighborhood V of $f(x)$, $f^{-1}(V)$ is a $({\tau }_1,{\tau }_2)\delta$-semineighborhood of x;
• (6)for each $x\in X$ and each ${\sigma }_1{\sigma }_2$-neighborhood V of $f(x)$, there exists a $({\tau }_1,{\tau }_2)\delta$-semineighborhood U of x such that $f(U)\subseteq V$;
• (7)$f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(B))\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(f^{-1}(B))$ for every subset B of Y;
• (8)$f^{-1}(V)\subseteq {\tau }_1\mathit{\text{-Cl}}({\tau }_2{\mathit{\text{-Int}}}_{\delta }(f^{-1}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y.

Definition 3.11

A subset A of a bitopological space $(X,{\tau }_1,{\tau }_2)$ is said to be $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-open if $A={\tau }_1{\text{-Int}}_{\delta }({\tau }_2{\text{-Int}}_{\delta }(A))$.

Lemma 3.12

Let $(X,{\tau }_1,{\tau }_2)$ be a bitopological space and $A,B\subseteq X$. If A is $({\tau }_1,{\tau }_2)\delta$-semiopen in $(X,{\tau }_1,{\tau }_2)$ and B is $({\tau }_1,{\tau }_2)\delta$-open in $(X,{\tau }_1,{\tau }_2)$, then $A\cap B$ is $({\tau }_{1_{{|}_B}},{\tau }_{2_{{|}_B}})\delta$-semiopen in $(B,{\tau }_{1_{{|}_B}},{\tau }_{2_{{|}_B}})$.

Proof

Suppose that A is $({\tau }_1,{\tau }_2)\delta$-semiopen in $(X,{\tau }_1,{\tau }_2)$ and B is $({\tau }_1,{\tau }_2)\delta$-open in $(X,{\tau }_1,{\tau }_2)$. Then $A\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(A))$ and $B={\tau }_1{\text{-Int}}_{\delta }({\tau }_2{\text{-Int}}_{\delta }(B))$. Therefore, we have $A\cap B\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(A))\cap B\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(A)\cap B)$ and hence

$$A\cap B\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(A)\cap B)\cap B={\tau }_1{\text{-Cl}}_B({\tau }_2{\text{-Int}}_{B_{\delta }}({\tau }_2{\text{-Int}}_{\delta }(A)\cap B))\subseteq {\tau }_1{\text{-Cl}}_B({\tau }_2{\text{-Int}}_{B_{\delta }}(A\cap B)).$$

This shows that $A\cap B$ is $({\tau }_{1_{{|}_B}},{\tau }_{2_{{|}_B}})\delta$-semiopen in $(B,{\tau }_{1_{{|}_B}},{\tau }_{2_{{|}_B}})$. □

Theorem 3.13

If a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous and G is $({\tau }_1,{\tau }_2)\delta$-open in $(X,{\tau }_1,{\tau }_2)$, then the restriction

$$F_{{|}_G}:(G,{\tau }_{1_{{|}_G}},{\tau }_{2_{{|}_G}})\to (Y,{\sigma }_1,{\sigma }_2)$$

is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

Proof

Let $x\in G$ and V be any ${\sigma }_1{\sigma }_2$-open subset of Y such that $F_{{|}_G}(x)\subseteq V$. Since F is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous and $F_{{|}_G}(x)=F(x)$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq V$. Put $U_0=U\cap G$, then by Lemma 3.12, $U_0$ is $({\tau }_{1_{{|}_G}},{\tau }_{2_{{|}_G}})\delta$-semiopen in $(G,{\tau }_{1_{{|}_G}},{\tau }_{2_{{|}_G}})$ containing x and $F_{{|}_G}(U_0)\subseteq V$. This shows that $F_{{|}_G}$ is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous. □

Theorem 3.14

If a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous and G is $({\tau }_1,{\tau }_2)\delta$-open in $(X,{\tau }_1,{\tau }_2)$, then the restriction

$$F_{{|}_G}:(G,{\tau }_{1_{{|}_G}},{\tau }_{2_{{|}_G}})\to (Y,{\sigma }_1,{\sigma }_2)$$

is lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

Proof

The proof is similar to that of Theorem 3.13. □

Lemma 3.15

Let $(X,{\tau }_1,{\tau }_2)$ be a bitopological space. If $A\subseteq U\subseteq X$, U is $({\tau }_1,{\tau }_2)\delta$-open in $(X,{\tau }_1,{\tau }_2)$ and A is $({\tau }_{1_{{|}_U}},{\tau }_{2_{{|}_U}})\delta$-semiopen in $(U,{\tau }_{1_{{|}_U}},{\tau }_{2_{{|}_U}})$, then A is $({\tau }_1,{\tau }_2)\delta$-semiopen in $(X,{\tau }_1,{\tau }_2)$.

Proof

Suppose that U is $({\tau }_1,{\tau }_2)\delta$-open in $(X,{\tau }_1,{\tau }_2)$ and A is $({\tau }_{1_{{|}_U}},{\tau }_{2_{{|}_U}})\delta$-semiopen in $(U,{\tau }_{1_{{|}_U}},{\tau }_{2_{{|}_U}})$. Since A is $({\tau }_{1_{{|}_U}},{\tau }_{2_{{|}_U}})\delta$-semiopen in $(U,{\tau }_{1_{{|}_U}},{\tau }_{2_{{|}_U}})$, we have $A\subseteq {\tau }_1{\text{-Cl}}_U({\tau }_2{\text{-Int}}_{U_{\delta }}(A))$ and hence

$$A\subseteq {\tau }_1{\text{-Cl}}_U({\tau }_2{\text{-Int}}_{U_{\delta }}(A))={\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{U_{\delta }}(A))\cap U\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{U_{\delta }}(A)\cap U)\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(A\cap U)).$$

Thus, A is $({\tau }_1,{\tau }_2)\delta$-semiopen in $(X,{\tau }_1,{\tau }_2)$. □

Theorem 3.16

A multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous if for each $x\in X$, there exists a $({\tau }_1,{\tau }_2)\delta$-open subset G of X containing x such that the restriction

$$F_{{|}_G}:(G,{\tau }_{1_{{|}_G}},{\tau }_{2_{{|}_G}})\to (Y,{\sigma }_1,{\sigma }_2)$$

is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

Proof

Let $x\in X$ and V be any ${\sigma }_1{\sigma }_2$-open set of Y such that $F(x)\subseteq V$. There exists a $({\tau }_1,{\tau }_2)\delta$-open subset G of X containing x such that $F_{{|}_G}$ is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous. Thus, there exists a $({\tau }_{1_{{|}_G}},{\tau }_{2_{{|}_G}})\delta$-semiopen subset $U_0$ of X containing x such that $F_{{|}_G}(U_0)\subseteq V$. By Lemma 3.15, $U_0$ is $({\tau }_1,{\tau }_2)\delta$-open in $(X,{\tau }_1,{\tau }_2)$ and $F(z)=F_{{|}_G}(z)$ for every $z\in U_0$. This shows that F is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous. □

Theorem 3.17

A multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous if for each $x\in X$, there exists a $({\tau }_1,{\tau }_2)\delta$-open subset G of X containing x such that the restriction

$$F_{{|}_G}:(G,{\tau }_{1_{{|}_G}},{\tau }_{2_{{|}_G}})\to (Y,{\sigma }_1,{\sigma }_2)$$

is lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

Proof

The proof is similar to that of Theorem 3.16. □

Recall that, for a multifunction $F:X\to Y$, the graph multifunction

$$G_F:X\to X\times Y$$

is defined as follows $G_F(x)=\{x\}\times F(x)$ for every $x\in X$ and the subset

$$\{\{x\}\times F(x)|x\in X\}\subseteq X\times Y$$

is called the multigraph of F and is denoted by $G(F)$ [30].

Lemma 3.18

[21] For a multifunction $F:X\to Y$, the following properties hold:

• (1)$G_F^{+}(A\times B)=A\cap F^{+}(B)$,
• (2)$G_F^{-}(A\times B)=A\cap F^{-}(B)$,

for any subsets $A\subseteq X$ and $B\subseteq Y$.

By $p_i$, we denote the product topology ${\tau }_i\times {\sigma }_i$ for $i=1,2$.

Theorem 3.19

If the graph multifunction of $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous, then F is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

Proof

Let $G_F:(X,{\tau }_1,{\tau }_2)\to (X\times Y,p_1,p_2)$ be upper $(p_1,p_2)\delta$-semicontinuous. Let $x\in X$ and V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $F(x)$. Since $X\times V$ is $p_1{p}_2$-open in $X\times Y$ and $G_F(x)\subseteq X\times V$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $G_F(U)\subseteq X\times V$. By Lemma 3.18, we have $U\subseteq G_F^{+}(X\times V)=F^{+}(V)$ and hence $F(U)\subseteq V$. This shows that F is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous. □

Theorem 3.20

A multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous provided that $G_F:(X,{\tau }_1,{\tau }_2)\to (X\times Y,p_1,p_2)$ is lower $(p_1,p_2)\delta$-semicontinuous.

Proof

Suppose that $G_F$ is lower $(p_1,p_2)\delta$-semicontinuous. Let $x\in X$ and V be any ${\sigma }_1{\sigma }_2$-open subset of Y such that $x\in F^{-}(V)$. Then $X\times V$ is $p_1{p}_2$-open in $X\times Y$ and

$$G_F(x)\cap (X\times V)=(\{x\}\times F(x))\cap (X\times V)=\{x\}\times (F(x)\cap V)\ne \varnothing .$$

Since $G_F$ is lower $(p_1,p_2)\delta$-semicontinuous, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $U\subseteq G_F^{-}(X\times V)$. By Lemma 3.18, we have $U\subseteq F^{-}(V)$ and hence F is lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous. □

For two multifunctions $F_1:X\to Y$ and $F_2:X\to Z$, the product multifunction $F_1\times F_2:X\to Y\times Z$ is defined as follows:

$$(F_1\times F_2)(x)=F_1(x)\times F_2(x)$$

for every $x\in X$.

Theorem 3.21

Let $F_1:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ and

$$F_2:(X,{\tau }_1,{\tau }_2)\to (Z,{\rho }_1,{\rho }_2)$$

be multifunctions. If $F_1\times F_2:(X,{\tau }_1,{\tau }_2)\to (Y\times Z,p_1,p_2)$ is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous, then $F_1$ and $F_2$ are upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

Proof

Let $x\in X$. Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $F(x)$ and W be any ${\rho }_1{\rho }_2$-open subset of Z containing $F(x)$. Then, we have $F_1(x)\times F_2(x)=(F_1\times F_2)(x)\subseteq V\times W$. Since $F_1\times F_2$ is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $(F_1\times F_2)(U)\subseteq V\times W$. Thus, $F_1(U)\subseteq V$ and $F_2(U)\subseteq W$. This shows that $F_1$ and $F_2$ are upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous. □

Theorem 3.22

Let $F_1:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ and

$$F_2:(X,{\tau }_1,{\tau }_2)\to (Z,{\rho }_1,{\rho }_2)$$

be multifunctions. If $F_1\times F_2:(X,{\tau }_1,{\tau }_2)\to (Y\times Z,p_1,p_2)$ is lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous, then $F_1$ and $F_2$ are lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

Proof

The proof is similar to that of Theorem 3.21. □

4. On characterizations of upper and lower almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunctions

In this section, we introduce the notions of upper and lower almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunctions. Moreover, some characterizations of upper and lower almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunctions are investigated. Furthermore, the relationships between $({\tau }_1,{\tau }_2)\delta$-semicontinuity and almost $({\tau }_1,{\tau }_2)\delta$-semicontinuity are discussed.

Definition 4.1

A multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is said to be:

• (1)upper almost $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicontinuous at a point $x\in X$ if for each ${\sigma }_1{\sigma }_2$-open subset V of Y such that $F(x)\subseteq V$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$;
• (2)lower almost $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicontinuous at a point $x\in X$ for each ${\sigma }_1{\sigma }_2$-open subset V of Y such that $F(x)\cap V\ne \varnothing$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that ${\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))\cap F(z)\ne \varnothing$ for every $z\in U$;
• (3)upper (resp. lower) almost $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicontinuous if F has this property at each point of X.

Remark 4.2

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following implication holds: upper $({\tau }_1,{\tau }_2)\delta$-semicontinuity ⇒ upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuity.

The converse of the implication is not true in general. We give an example for the implication as follows.

Example 4.3

Let $X=\{a,b,c,d\}$ with topologies ${\tau }_1=\{\varnothing ,\{a\},\{b\},\{a,b\},X\}$ and ${\tau }_2=\{\varnothing ,\{b\},X\}$. Let $Y=\{-2,-1,0,1,2\}$ with topologies

$${\sigma }_1=\{\varnothing ,\{-2,-1,2\},\{0,1\},Y\}$$

and ${\sigma }_2=\{\varnothing ,\{-2,-1,2\},Y\}$. A multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is defined as follows:

$$F(x)=\{\begin{array}{cc}\{-2,-1\},\hfill & \text{if }x=a,\hfill \\ \{2\},\hfill & \text{if }x=b,\hfill \\ \{0,1\},\hfill & \text{if }x=c,\hfill \\ \{-1,2\},\hfill & \text{if }x=d.\hfill \end{array}$$

Then F is upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous but F is not upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

Definition 4.4

A subset A of a bitopological space $(X,{\tau }_1,{\tau }_2)$ is said to be $({\mathit{\tau }}_1,{\mathit{\tau }}_2)s$-open if $A\subseteq {\tau }_1{\tau }_2\text{-Cl}({\tau }_1{\tau }_2\text{-Int}(A))$. The complement of a $({\tau }_1,{\tau }_2)s$-open set is said to be $({\mathit{\tau }}_1,{\mathit{\tau }}_2)s$-closed. The intersection of all $({\tau }_1,{\tau }_2)s$-closed sets of X containing A is called $({\mathit{\tau }}_1,{\mathit{\tau }}_2)s$-closure of A and is denoted by $({\tau }_1,{\tau }_2)\text{-sCl}(A)$. The union of all $({\tau }_1,{\tau }_2)s$-open sets of X contained in A is called $({\mathit{\tau }}_1,{\mathit{\tau }}_2)s$-interior of A and is denoted by $({\tau }_1,{\tau }_2)\text{-sInt}(A)$.

Lemma 4.5

For a subset A of a bitopological space $(X,{\tau }_1,{\tau }_2)$, the following properties hold:

• (1)$({\tau }_1,{\tau }_2)\mathit{\text{-sCl}}(A)={\tau }_1{\tau }_2\mathit{\text{-Int}}({\tau }_1{\tau }_2\mathit{\text{-Cl}}(A))\cup A$.
• (2)If A is ${\tau }_1{\tau }_2$-open in X, then $({\tau }_1,{\tau }_2)\mathit{\text{-sCl}}(A)={\tau }_1{\tau }_2\mathit{\text{-Int}}({\tau }_1{\tau }_2\mathit{\text{-Cl}}(A))$.

Proof

(1) Since $({\tau }_1,{\tau }_2)\text{-sCl}(A)$ is $({\tau }_1,{\tau }_2)s$-closed, we have

$${\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}(({\tau }_1,{\tau }_2)\text{-sCl}(A)))\subseteq ({\tau }_1,{\tau }_2)\text{-sCl}(A)$$

and hence ${\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}(A))\subseteq ({\tau }_1,{\tau }_2)\text{-sCl}(A)$. Thus,

$${\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}(A))\cup A\subseteq ({\tau }_1,{\tau }_2)\text{-sCl}(A).$$

To establish the opposite inclusion, we observe that

$${\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}({\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}(A))\cup A))\subseteq {\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}({\tau }_1{\tau }_2\text{-Cl}(A)\cup A))={\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}(A))\subseteq {\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}(A))\cup A$$

and hence ${\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}(A))\cup A$ is $({\tau }_1,{\tau }_2)s$-closed. Thus, $({\tau }_1,{\tau }_2)\text{-sCl}(A)\subseteq {\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}(A))\cup A$. Consequently, we obtain

$$({\tau }_1,{\tau }_2)\text{-sCl}(A)={\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}(A))\cup A.$$

(2) Suppose that A is a ${\tau }_1{\tau }_2$-open set. Then $A\subseteq {\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}(A))$ and by (1),

$$({\tau }_1,{\tau }_2)\text{-sCl}(A)={\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}(A))\cup A={\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-}\text{Cl}(A)).$$

 □

Definition 4.6

A subset A of a bitopological space $(X,{\tau }_1,{\tau }_2)$ is said to be:

• (i)$({\mathit{\tau }}_1,{\mathit{\tau }}_2)r$-open if $A={\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}(A))$ [33];
• (ii)$({\mathit{\tau }}_1,{\mathit{\tau }}_2)r$-closed if $A={\tau }_1{\tau }_2\text{-Cl}({\tau }_1{\tau }_2\text{-Int}(A))$ [33];
• (iii)$({\mathit{\tau }}_1,{\mathit{\tau }}_2)p$-open if $A\subseteq {\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}(A))$;
• (iv)$({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\beta }$-open if $A\subseteq {\tau }_1{\tau }_2\text{-Cl}({\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}(A)))$.

Theorem 4.7

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous at $x\in X$;
• (2)$x\in ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y containing $F(x)$;
• (3)$x\in ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{+}(({\sigma }_1,{\sigma }_2)\mathit{\text{-sCl}}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y containing $F(x)$;
• (4)$x\in ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{+}(V))$ for every $({\sigma }_1,{\sigma }_2)r$-open subset V of Y containing $F(x)$;
• (5)for each $({\sigma }_1,{\sigma }_2)r$-open subset V of Y containing $F(x)$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq (V)$.

Proof

$(1)\Rightarrow (2)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $F(x)$. Since F is upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous at x, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$. Thus, $x\in U\subseteq F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V)))$ and hence $x\in ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))$.

$(2)\Rightarrow (3)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $F(x)$. By (2), we have $x\in ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))$ and by Lemma 4.5(2),

$$x\in ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(({\sigma }_1,{\sigma }_2)\text{-sCl}(V))).$$

$(3)\Rightarrow (4)$: Let V be any $({\sigma }_1,{\sigma }_2)r$-open subset of Y containing $F(x)$. Then V is ${\sigma }_1{\sigma }_2$-open and by (3), we have

$$x\in ({\tau }_1,{\tau }_2)\delta \text{-}\text{sInt}(F^{+}(({\sigma }_1,{\sigma }_2)\text{-}\text{sCl}(V))).$$

By Lemma 4.5(2),

$$x\in ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))=({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(V)).$$

$(4)\Rightarrow (5)$: Let V be any $({\sigma }_1,{\sigma }_2)r$-open subset of Y containing $F(x)$. By (4), we have $x\in ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(V))$ and there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $U\subseteq F^{+}(V)$; hence $F(U)\subseteq V$.

$(5)\Rightarrow (1)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $F(x)$. Since

$${\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$$

is $({\sigma }_1,{\sigma }_2)r$-open and by (5), there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$. This shows that F is upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous at x. □

Theorem 4.8

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is lower almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous at $x\in X$;
• (2)$x\in ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y such that $F(x)\cap V\ne \varnothing$;
• (3)$x\in ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{-}(({\sigma }_1,{\sigma }_2)\mathit{\text{-sCl}}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y such that $F(x)\cap V\ne \varnothing$;
• (4)$x\in ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{-}(V))$ for every $({\sigma }_1,{\sigma }_2)r$-open subset V of Y such that $F(x)\cap V\ne \varnothing$;
• (5)for each $({\sigma }_1,{\sigma }_2)r$-open subset V of Y such that $F(x)\cap V\ne \varnothing$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $U\subseteq F^{-}(V)$.

Proof

The proof is similar to that of Theorem 4.7. □

Definition 4.9

A function $f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is said to be almost $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicontinuous at a point $x\in X$ if for each ${\sigma }_1{\sigma }_2$-open subset V of Y containing $f(x)$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $f(U)\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$. If f has this property at each point of X, then f is said to be almost $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicontinuous.

Remark 4.10

For a function $f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following implication holds:

$$({\tau }_1,{\tau }_2)\delta \text{-semicontinuity}\Rightarrow \text{ almost }({\tau }_1,{\tau }_2)\delta \text{-semicontinuity}.$$

The converse of the implication is not true in general. We give an example for the implication as follows.

Example 4.11

Let $X=\{a,b,c,d\}$ with topologies

$${\tau }_1=\{\varnothing ,\{a\},\{b\},\{a,b\},X\}$$

and ${\tau }_2=\{\varnothing ,\{b\},X\}$. Let $Y=\{-1,0,1,2\}$ with topologies

$${\sigma }_1=\{\varnothing ,\{-1,0\},Y\}$$

and ${\sigma }_2=\{\varnothing ,\{-1,0\},\{1,2\},Y\}$. Define a function

$$f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$$

as follows: $f(a)=-2$, $f(b)=-1$, $f(c)=1$ and $f(d)=0$. Then f is almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous but f is not $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

Corollary 4.12

For a function $f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)f is almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous at $x\in X$;
• (2)$x\in ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y containing $f(x)$;
• (3)$x\in ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(f^{-1}(({\sigma }_1,{\sigma }_2)\mathit{\text{-sCl}}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y containing $f(x)$;
• (4)$x\in ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(f^{-1}(V))$ for every $({\sigma }_1,{\sigma }_2)r$-open subset V of Y containing $f(x)$;
• (5)for each $({\sigma }_1,{\sigma }_2)r$-open subset V of Y containing $f(x)$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $f(U)\subseteq (V)$.

Theorem 4.13

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)$F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V)))$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (3)$F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(K)))$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (4)$F^{+}(V)$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X for every $({\sigma }_1,{\sigma }_2)r$-open subset V of Y;
• (5)$F^{-}(K)$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X for every $({\sigma }_1,{\sigma }_2)r$-closed subset K of Y;
• (6)for each $x\in X$ and each ${\sigma }_1{\sigma }_2$-open subset V of Y containing $F(x)$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq {\sigma }_1{\sigma }_2\mathit{\text{-sCl}}(V)$;
• (7)$F^{+}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{+}(({\sigma }_1,{\sigma }_2)\mathit{\text{-sCl}}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (8)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}(({\sigma }_1,{\sigma }_2)\mathit{\text{-sInt}}(K)))\subseteq F^{-}(K)$ for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (9)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(K))))\subseteq F^{-}(K)$ for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (10)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}(V))\subseteq F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)\beta$-open subset V of Y;
• (11)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}(V))\subseteq F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)s$-open subset V of Y;
• (12)$F^{+}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y.

Proof

$(1)\Rightarrow (2)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y and

$$x\in F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))).$$

Then $F(x)\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$. Since F is upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$ and hence

$$x\in U\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(U))\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))).$$

Therefore,

$$F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-}\text{Cl}(V))))).$$

Consequently, we obtain $F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V)))$ is $({\tau }_1,{\tau }_2)\delta$- semiopen in X.

$(2)\Rightarrow (1)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y and $x\in F^{+}(V)$. By (2), we have $F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V)))$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X. Put

$$U=F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))).$$

Then U is a $({\tau }_1,{\tau }_2)\delta$-semiopen subset of X containing x such that

$$F(U)\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V)).$$

This shows that F is lower almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

$(2)\iff (3)$: Let K be any ${\sigma }_1{\sigma }_2$-closed subset of Y. Then $Y-K$ is ${\sigma }_1{\sigma }_2$-open in Y and by (2), $F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(Y-K)))$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X. Since ${\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(Y-K))=Y-{\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}(K))$, it follows that

$$F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(Y-K)))=X-F^{-}({\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}(K)))$$

and hence $F^{-}({\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}(K)))$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X. The converse implication is proved by analogy.

$(2)\Rightarrow (4)$: Let V be any $({\sigma }_1,{\sigma }_2)r$-open subset of Y. Then V is ${\sigma }_1{\sigma }_2$-open in Y and by (2), we have $F^{+}(V)=F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X.

$(4)\Rightarrow (2)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y. Then

$${\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-}\text{Cl}(V))$$

is $({\sigma }_1,{\sigma }_2)r$-open in Y. By (4), $F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-}\text{Cl}(V))$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X.

$(3)\Rightarrow (5)$: Let K be any $({\sigma }_1,{\sigma }_2)r$-closed subset of Y. Then K is ${\sigma }_1{\sigma }_2$-closed in Y and by (3), $F^{-}(K)=F^{-}({\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}(K)))$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X.

$(5)\Rightarrow (3)$: Let K be any ${\sigma }_1{\sigma }_2$-closed subset of Y. Then

$${\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-}\text{Int}(K))$$

is $({\sigma }_1,{\sigma }_2)r$-closed in Y. By (5), $F^{-}({\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}(K)))$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X.

$(1)\iff (6)$: Let $x\in X$ and V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $F(x)$. By (1), there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$ and by Lemma 4.5(2), $F(U)\subseteq {\sigma }_1{\sigma }_2\text{-sCl}(V)$. The converse implication is obtained similarly.

$(5)\Rightarrow (7)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y and $x\in F^{+}(V)$. Then, we have $F(x)\subseteq V\subseteq {\sigma }_1{\sigma }_2\text{-sCl}(V)$ and hence

$$x\in F^{+}({\sigma }_1{\sigma }_2\text{-sCl}(V))=X-F^{-}(Y-{\sigma }_1{\sigma }_2\text{-sCl}(V)).$$

Since $Y-{\sigma }_1{\sigma }_2\text{-sCl}(V)$ is $({\sigma }_1,{\sigma }_2)r$-closed in Y and by (5),

$$F^{-}(Y-{\sigma }_1{\sigma }_2\text{-sCl}(V))$$

is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X. Thus, $F^{+}({\sigma }_1{\sigma }_2\text{-sCl}(V))$ is $({\tau }_1,{\tau }_2)\delta$-semiopen and hence $x\in ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(({\sigma }_1,{\sigma }_2)\text{-sCl}(V)))$. Consequently, we obtain $F^{+}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(({\sigma }_1,{\sigma }_2)\text{-sCl}(V)))$.

$(7)\Rightarrow (8)$: Let K be any ${\sigma }_1{\sigma }_2$-closed subset of Y. Then $Y-K$ is ${\sigma }_1{\sigma }_2$-open and by (7), we have

$$X-F^{-}(K)=F^{+}(Y-K)\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(({\sigma }_1,{\sigma }_2)\text{-sCl}(Y-K)))=({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(Y-({\sigma }_1,{\sigma }_2)\text{-sInt}(K)))=({\tau }_1,{\tau }_2)\delta \text{-sInt}(X-F^{-}(({\sigma }_1,{\sigma }_2)\text{-sInt}(K)))=X-({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}(({\sigma }_1,{\sigma }_2)\text{-sInt}(K)))$$

and hence $({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}(({\sigma }_1,{\sigma }_2)\text{-sInt}(K)))\subseteq F^{-}(K)$.

$(8)\Rightarrow (9)$: Let K be any ${\sigma }_1{\sigma }_2$-closed subset of Y. Since $({\sigma }_1,{\sigma }_2)\text{-sInt}(K)={\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Cl}(K))$ and by (8),

$$({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}(K))))\subseteq F^{-}(K).$$

$(9)\Rightarrow (5)$: Let K be any $({\sigma }_1,{\sigma }_2)r$-closed subset of Y. Then K is ${\sigma }_1{\sigma }_2$-closed in Y and by (9),

$$({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}(K))=({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-}\text{Cl}({\sigma }_1{\sigma }_2\text{-Int}(K))))\subseteq F^{-}(K).$$

This shows that $F^{-}(K)$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X.

$(5)\Rightarrow (10)$: Let V be any $({\sigma }_1,{\sigma }_2)\beta$-open subset of Y. Then ${\sigma }_1{\sigma }_2\text{-Cl}(V)$ is $({\sigma }_1,{\sigma }_2)r$-closed in Y. By (5), we have $F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(V))$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed and hence $({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}(V))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(V))$.

$(10)\Rightarrow (11)$: Since every $({\sigma }_1,{\sigma }_2)s$-open set is $({\sigma }_1,{\sigma }_2)\beta$-open, the proof is obvious.

$(11)\Rightarrow (12)$: Let V be any $({\sigma }_1,{\sigma }_2)p$-open subset of Y. Then, we have the inclusion $V\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$ and ${\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}(Y-V))\subseteq Y-V$. Moreover, since the set ${\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}(Y-V))$ is $({\sigma }_1,{\sigma }_2)s$-open and by (11),

$$X-({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))=({\tau }_1,{\tau }_2)\delta \text{-sCl}(X-F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))=({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}(Y-{\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))=({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}(Y-V))))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}(Y-V))\subseteq F^{-}(Y-V)=X-F^{+}(V).$$

Thus, $F^{+}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))$.

$(12)\Rightarrow (4)$: Let V be any $({\sigma }_1,{\sigma }_2)r$-open subset of Y. Then V is $({\sigma }_1,{\sigma }_2)p$-open in Y. By (12),

$$F^{+}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))=({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(V))$$

and hence $F^{+}(V)$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X. □

Theorem 4.14

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is lower almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)$F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V)))$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (3)$F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(K)))$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (4)$F^{-}(V)$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X for every $({\sigma }_1,{\sigma }_2)r$-open subset V of Y;
• (5)$F^{+}(K)$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X for every $({\sigma }_1,{\sigma }_2)r$-closed subset K of Y;
• (6)for each $x\in X$ and each ${\sigma }_1{\sigma }_2$-open subset V of Y such that $F(x)\cap V\ne \varnothing$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that ${\sigma }_1{\sigma }_2\mathit{\text{-sCl}}(V)\cap F(z)$ for every $z\in U$;
• (7)$F^{-}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{-}(({\sigma }_1,{\sigma }_2)\mathit{\text{-sCl}}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (8)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}(({\sigma }_1,{\sigma }_2)\mathit{\text{-sInt}}(K)))\subseteq F^{+}(K)$ for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (9)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(K))))\subseteq F^{+}(K)$ for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (10)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}(V))\subseteq F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)\beta$-open subset V of Y;
• (11)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}(V))\subseteq F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)s$-open subset V of Y;
• (12)$F^{-}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y.

Proof

The proof is similar to that of Theorem 4.13. □

Corollary 4.15

For a function $f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)f is almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)$f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V)))$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (3)$f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(K)))$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (4)$f^{-1}(V)$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X for every $({\sigma }_1,{\sigma }_2)r$-open subset V of Y;
• (5)$f^{-1}(K)$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X for every $({\sigma }_1,{\sigma }_2)r$-closed subset K of Y;
• (6)for each $x\in X$ and each ${\sigma }_1{\sigma }_2$-open subset V of Y containing $f(x)$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $f(U)\subseteq {\sigma }_1{\sigma }_2\mathit{\text{-sCl}}(V)$;
• (7)$f^{-1}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(f^{-1}(({\sigma }_1,{\sigma }_2)\mathit{\text{-sCl}}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (8)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}(({\sigma }_1,{\sigma }_2)\mathit{\text{-sInt}}(K)))\subseteq f^{-1}(K)$ for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (9)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(K))))\subseteq f^{-1}(K)$ for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (10)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}(V))\subseteq f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)\beta$-open subset V of Y;
• (11)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}(V))\subseteq f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)s$-open subset V of Y;
• (12)$f^{-1}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y.

Lemma 4.16

[15] Let $\{X_{\gamma }|\gamma \in \mathrm{\Gamma }\}$ be any family of topological spaces and $U_{{\gamma }_k}$ be a non-empty subset of $X_{{\gamma }_k}$ for each $k=1,2,...,n$. Then

$$\underset{k=1}{\prod ^n}{U}_{{\gamma }_k}\times \prod _{\gamma \ne {\gamma }_k}{X}_{\gamma }$$

is δ-semiopen subset of $\prod _{\gamma \in \mathrm{\Gamma }}{X}_{\gamma }$ if and only if $U_{{\gamma }_k}$ is δ-semiopen in $X_{{\gamma }_k}$ for each $k=1,2,...,n$.

Let $\{(X_{\gamma },{\tau }_1(\gamma ),{\tau }_2(\gamma ))|\gamma \in \mathrm{\Gamma }\}$ be a family of bitopological spaces. Let $(X^{⁎},{\tau }_1^{⁎},{\tau }_2^{⁎})$ be the product space, where $X^{⁎}=\prod _{\gamma \in \mathrm{\Gamma }}{X}_{\gamma }$ and ${\tau }_i^{⁎}$ denotes the product topology of $\{{\tau }_i(\gamma )|\gamma \in \mathrm{\Gamma }\}$ for $i=1,2$.

Lemma 4.17

Let $\{(X_{\gamma },{\tau }_1(\gamma ),{\tau }_2(\gamma ))|\gamma \in \mathrm{\Gamma }\}$ be a family of bitopological spaces. Let $A_{{\gamma }_k}$ be a non-empty subset of $X_{{\gamma }_k}$ for $k=1,2,...,n$. Then

$$\underset{k=1}{\prod ^n}{A}_{{\gamma }_k}\times \prod _{\gamma \ne {\gamma }_k}{X}_{\gamma }$$

is $({\tau }_1^{⁎},{\tau }_2^{⁎})\delta$-semiopen if and only if $A_{{\gamma }_k}$ is $({\tau }_1({\gamma }_k),{\tau }_2({\gamma }_k))\delta$-semiopen in $X_{{\gamma }_k}$ for each $k=1,2,...,n$.

Proof

The proof is similar to that of Lemma 4.16. □

Let $\{(X_{\gamma },{\tau }_1(\gamma ),{\tau }_2(\gamma ))|\gamma \in \mathrm{\Gamma }\}$ and $\{(Y_{\gamma },{\sigma }_1(\gamma ),{\sigma }_2(\gamma ))|\gamma \in \gamma \}$ be two arbitrary families of bitopological spaces with the same set of indices. Let

$$F_{\gamma }:(X_{\gamma },{\tau }_1(\gamma ),{\tau }_2(\gamma ))\to (Y_{\gamma },{\sigma }_1(\gamma ),{\sigma }_2(\gamma ))$$

be a multifunction for each $\gamma \in \mathrm{\Gamma }$. Let $F^{⁎}:(X^{⁎},{\tau }_1^{⁎},{\tau }_2^{⁎})\to (Y^{⁎},{\sigma }_1^{⁎},{\sigma }_2^{⁎})$ be the product multifunction defined by $F^{⁎}(x^{⁎})=\prod _{\gamma \in \mathrm{\Gamma }}\{F_{\gamma }(x_{\gamma })\}$ for each

$$x^{⁎}=\{x_{\gamma }\}\in \prod _{\gamma \in \mathrm{\Gamma }}{X}_{\gamma },$$

where ${\tau }_i^{⁎}$ and ${\sigma }_i^{⁎}$ denote the product topologies for $i=1,2$.

Theorem 4.18

If $F^{⁎}:(X^{⁎},{\tau }_1^{⁎},{\tau }_2^{⁎})\to (Y^{⁎},{\sigma }_1^{⁎},{\sigma }_2^{⁎})$ is upper almost $({\tau }_1^{⁎},{\tau }_2^{⁎})\delta$-semicontinuous, then $F_{\gamma }:(X_{\gamma },{\tau }_1(\gamma ),{\tau }_2(\gamma ))\to (Y_{\gamma },{\sigma }_1(\gamma ),{\sigma }_2(\gamma ))$ is upper almost $({\tau }_1(\gamma ),{\tau }_2(\gamma ))\delta$-semicontinuous for each $\gamma \in \mathrm{\Gamma }$.

Proof

Let $V_{\gamma }$ be any $({\sigma }_1(\gamma ),{\sigma }_2(\gamma ))r$-open subset of $Y_{\gamma }$. Since $F^{⁎}$ is upper almost $({\tau }_1^{⁎},{\tau }_2^{⁎})\delta$-continuous, $F^{⁎+}(V_{\gamma }\times \prod _{\gamma \ne {\gamma }_0}{Y}_{{\gamma }_0})=F_{\gamma }^{+}(V_{\gamma })\times \prod _{\gamma \ne {\gamma }_0}{Y}_{{\gamma }_0}$ is $({\tau }_1^{⁎},{\tau }_2^{⁎})\delta$-semiopen in $X^{⁎}$ and by Lemma 4.17, $F_{\gamma }^{+}(V_{\gamma })$ is $({\tau }_1(\gamma ),{\tau }_2(\gamma ))\delta$-semiopen in $X_{\gamma }$. This shows that $F_{\gamma }$ is upper almost $({\tau }_1(\gamma ),{\tau }_2(\gamma ))\delta$-semicontinuous. □

Theorem 4.19

If $F^{⁎}:(X^{⁎},{\tau }_1^{⁎},{\tau }_2^{⁎})\to (Y^{⁎},{\sigma }_1^{⁎},{\sigma }_2^{⁎})$ is lower almost $({\tau }_1^{⁎},{\tau }_2^{⁎})\delta$-semicontinuous, then $F_{\gamma }:(X_{\gamma },{\tau }_1(\gamma ),{\tau }_2(\gamma ))\to (Y_{\gamma },{\sigma }_1(\gamma ),{\sigma }_2(\gamma ))$ is lower almost $({\tau }_1(\gamma ),{\tau }_2(\gamma ))\delta$-semicontinuous for each $\gamma \in \mathrm{\Gamma }$.

Proof

The proof is similar to that of Theorem 4.18. □

Recall that a net $(x_{\alpha })$ in a topological space $(X,\tau )$ is called eventually in the set $U\subseteq X$ if there exists an index ${\alpha }_0\in \mathrm{\nabla }$ such that $x_{\alpha }\in U$ for all $\alpha \ge {\alpha }_0$.

Definition 4.20

Let $(X,{\tau }_1,{\tau }_2)$ be a bitopological space, $x\in X$ and let $(x_{\alpha })$ be a net in X. We say that the net $(x_{\alpha })$ $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semiconverges to x if for each $({\tau }_1,{\tau }_2)\delta$-semiopen set G containing x, there exists an index ${\alpha }_0\in \mathrm{\nabla }$ such that $x_{\alpha }\in G$ for each $\alpha \ge {\alpha }_0$.

Theorem 4.21

If $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is an upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunction, then for each $x\in X$ and for each net $(x_{\alpha })$ which $({\tau }_1,{\tau }_2)\delta$-semiconverges to x in X and for each ${\sigma }_1{\sigma }_2$-open subset V of Y such that $x\in F^{+}(V)$, the net $(x_{\alpha })$ is eventually in $F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V)))$.

Proof

Let $(x_{\alpha })$ be a net which $({\tau }_1,{\tau }_2)\delta$-semiconverges to x in X and let V be any ${\sigma }_1{\sigma }_2$-open subset of Y such that $x\in F^{+}(V)$. Since F is upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $U\subseteq F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V)))$. Since $(x_{\alpha })$ $({\tau }_1,{\tau }_2)\delta$-semiconverges to x, there exists an index ${\alpha }_0\in \mathrm{\nabla }$ such that $x_{\alpha }\in U$ for all $\alpha \ge {\alpha }_0$. Therefore, $x_{\alpha }\in U\subseteq F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V)))$ for all $\alpha \ge {\alpha }_0$. Thus, the net $(x_{\alpha })$ is eventually in $F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-}\text{Cl}(V)))$. □

Theorem 4.22

If $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is a lower almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunction, then for each $x\in X$ and for each net $(x_{\alpha })$ which $({\tau }_1,{\tau }_2)\delta$-semiconverges to x in X and for each ${\sigma }_1{\sigma }_2$-open subset V of Y such that $x\in F^{-}(V)$, the net $(x_{\alpha })$ is eventually in $F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V)))$.

Proof

The proof is similar to that of Theorem 4.21. □

In the following $(D,>)$ is a directed set, $(F_{\gamma })$ is a net of multifunction

$$F_{\gamma }:X\to Y$$

for every $\gamma \in D$ and F is a multifunction from X into Y.

Definition 4.23

Let ${(F_{\gamma })}_{\gamma \in D}$ be a net of multifunction from $(X,{\tau }_1,{\tau }_2)$ into $(Y,{\sigma }_1,{\sigma }_2)$. A multifunction $F^{⊛}:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is defined as follows: for each $x\in X$, $F^{⊛}(x)=\{y\in Y|\text{for each}{\sigma }_1{\sigma }_2\text{-open neighbor}\text{hood}V\text{of}y\text{and each }\lambda \in D,\text{there exists}\gamma \in D\text{such that}\gamma >\lambda \text{and}V\cap F_{\gamma }\ne \varnothing \}$ is called the upper bitopological limit of the net ${(F_{\gamma })}_{\gamma \in D}$.

A net ${(F_{\gamma })}_{\gamma \in D}$ is said to be equally upper almost $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicontinuous at $x_0\in X$ if for every ${\tau }_1{\tau }_2$-open set V containing $F_{\gamma }(x_0)$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen set U containing $x_0$ such that $F_{\gamma }(U)\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$ for all $\gamma \in D$.

Definition 4.24

[2] A bitopological space $(X,{\tau }_1,{\tau }_2)$ is said to be ${\mathit{\tau }}_1{\mathit{\tau }}_2$-compact if every cover of X by ${\tau }_1{\tau }_2$-open subsets of X has a finite subcover.

Theorem 4.25

Let ${(F_{\gamma })}_{\gamma \in D}$ be a net of multifunction from a bitopological space $(X,{\tau }_1,{\tau }_2)$ into a ${\sigma }_1{\sigma }_2$-compact space $(Y,{\sigma }_1,{\sigma }_2)$. If the following are satisfied:

• (1)$\cap \{(Y-F_{\lambda }(x))|\lambda >\gamma \}$ is ${\sigma }_1{\sigma }_2$-open in Y for each $\gamma \in D$ and each $x\in X$,
• (2)${(F_{\gamma })}_{\gamma \in D}$ is equally upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous on X,

then $F^{⊛}$ is upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous on X.

Proof

It is known that $F^{⊛}(x)=\cap \{{\sigma }_1{\sigma }_2\text{-Cl}(\cup \{F_{\lambda }(x)|\lambda >\gamma \}):\gamma \in D\}$ and by (1), $F^{⊛}(x)=\cap \{(\cup \{F_{\lambda }(x)|\lambda >\gamma \}):\gamma \in D\}$. Since the net

$${(\cup \{F_{\lambda }(x)|\lambda >\gamma \})}_{\gamma }\in D$$

is a family of ${\sigma }_1{\sigma }_2$-closed sets having the finite intersection property and Y is ${\sigma }_1{\sigma }_2$-compact, it follows that $F^{⊛}(x)\ne \varnothing$ for each $x\in X$. Now, let $x_0\in X$ and let V be any ${\sigma }_1{\sigma }_2$-open subset of Y such that $V\ne Y$ and $F^{⊛}(x_0)\subseteq V$. Then $F^{⊛}(x_0)\cap (Y-V)=\varnothing$, $F^{⊛}(x_0)\ne \varnothing$ and $Y-V\ne \varnothing$. It results that

$$\cap \{(\cup \{F_{\lambda }(x)|\lambda >\gamma \}):\gamma \in D\}\cap (Y-V)=\varnothing$$

and hence $\cap \{(\cup \{F_{\lambda }(x_0)\cap (Y-V)|\lambda >\gamma \}):\gamma \in D\}=\varnothing$. Since Y is ${\sigma }_1{\sigma }_2$-compact and the family $\{(\cup \{F_{\lambda }(x_0)\cap (Y-V)|\lambda >\gamma \}):\gamma \in D\}$ is a family of ${\sigma }_1{\sigma }_2$-closed sets with the empty intersection, there exists $\gamma \in D$ such that for each $\lambda \in D$ with $\lambda >\gamma$ we have $F_{\lambda }(x_0)\cap (Y-V)=\varnothing$; hence $F_{\lambda }(x_0)\subseteq V$. Since the net ${(F_{\gamma })}_{\gamma \in D}$ is equally upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous on X, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing $x_0$ such that $F_{\lambda }(U)\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$ for each $\lambda >\gamma$; hence $F_{\lambda }(x)\cap (Y-{\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V)))=\varnothing$ for each $x\in U$. Then, we have $\cup \{F_{\lambda }(x)\cap (Y-{\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V)))|\lambda >\gamma \}=\varnothing$; hence $\cap \{(\cup \{F_{\lambda }(x)|\lambda >\gamma \}):\gamma \in D\}\cap (Y-{\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V)))=\varnothing$. This implies that $F^{⊛}(U)\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$. If $V=Y$, then it is clear that for each $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing $x_0$ we have $F^{⊛}(U)\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$. Thus, $F^{⊛}$ is upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous at $x_0$. Since $x_0$ is arbitrary, the proof completes. □

5. On characterizations of upper and lower weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunctions

In this section, we introduce the notions of upper and lower weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunctions. Moreover, several characterizations of upper and lower weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous multifunctions are investigated. Furthermore, the relationships between almost $({\tau }_1,{\tau }_2)\delta$-semicontinuity and weak $({\tau }_1,{\tau }_2)\delta$-semicontinuity are discussed.

Definition 5.1

A multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is said to be:

• (1)upper weakly $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicontinuous at a point $x\in X$ if for each ${\sigma }_1{\sigma }_2$-open subset V of Y such that $F(x)\subseteq V$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq {\sigma }_1{\sigma }_2\text{-Cl}(V)$;
• (2)lower weakly $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicontinuous at a point $x\in X$ if for each ${\sigma }_1{\sigma }_2$-open subset V of Y such that $F(x)\cap V\ne \varnothing$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that ${\sigma }_1{\sigma }_2\text{-Cl}(V)\cap F(z)\ne \varnothing$ for every $z\in U$;
• (3)upper (resp. lower) weakly $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicontinuous if F has this property at each point of X.

Remark 5.2

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following implication holds: upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuity ⇒ upper weak $({\tau }_1,{\tau }_2)\delta$-semicontinuity.

The converse of the implication is not true in general. We give an example for the implication as follows.

Example 5.3

Let $X=\{a,b,c,d\}$ with topologies

$${\tau }_1=\{\varnothing ,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\},\{a,c,d\},X\}$$

and ${\tau }_2=\{\varnothing ,\{a\},\{c\},\{a,b\},\{a,c\},\{a,b,c\},\{a,c,d\},X\}$. Let

$$Y=\{-2,-1,0,1\}$$

with topologies

$${\sigma }_1=\{\varnothing ,\{-2\},\{0\},\{-2,-1\},\{-2,0\},\{-2,-1,0\},\{-2,0,1\},Y\}$$

and

$${\sigma }_2=\{\varnothing ,\{-2\},\{0\},\{-1\},\{-2,-1\},\{-2,0\},\{-1,0\},\{-2,-1,0\},\{-2,0,1\},Y\}.$$

A multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is defined as follows:

$$F(x)=\{\begin{array}{cc}\{-2,-1\},\hfill & \text{if }x=a,\hfill \\ \{1\},\hfill & \text{if }x=b,\hfill \\ \{-2,-1,0\},\hfill & \text{if }x=c,\hfill \\ \{-2,0,1\},\hfill & \text{if }x=d.\hfill \end{array}$$

Then F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous but F is not upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

Theorem 5.4

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous at $x\in X$;
• (2)$x\in ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y containing $F(x)$;
• (3)$x\in {\tau }_1\mathit{\text{-Cl}}({\tau }_2{\mathit{\text{-Int}}}_{\delta }(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y containing $F(x)$.

Proof

$(1)\Rightarrow (2)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $F(x)$. Since F is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous at x, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq {\sigma }_1{\sigma }_2\text{-Cl}(V)$. Thus, $x\in U\subseteq F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V))$. Since U is $({\tau }_1,{\tau }_2)\delta$-semiopen, we have

$$x\in U=({\tau }_1,{\tau }_2)\delta \text{-sInt}(U)\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V))).$$

$(2)\Rightarrow (3)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $F(x)$. By (2) and Lemma 3.2(2),

$$x\in ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V)))={\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V))))\cap F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V))\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V)))).$$

$(3)\Rightarrow (1)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $F(x)$. By (3), we have $x\in {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V))))$. Since $x\in F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V))$ and by Lemma 3.2(2), $x\in ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V)))$. Put

$$U=({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V))).$$

Then U is a $({\tau }_1,{\tau }_2)\delta$-semiopen subset of X containing x such that $F(U)\subseteq {\sigma }_1{\sigma }_2\text{-Cl}(V)$. This shows that F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous at x. □

Theorem 5.5

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is lower weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous at $x\in X$;
• (2)$x\in ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y such that $F(x)\cap V\ne \varnothing$;
• (3)$x\in {\tau }_1\mathit{\text{-Cl}}({\tau }_2{\mathit{\text{-Int}}}_{\delta }(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y such that $F(x)\cap V\ne \varnothing$.

Proof

The proof is similar to that of Theorem 5.4. □

Definition 5.6

A function $f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is said to be weakly $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicontinuous at a point $x\in X$ if for each ${\sigma }_1{\sigma }_2$-open subset V of Y such containing $f(x)$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $f(U)\subseteq {\sigma }_1{\sigma }_2\text{-Cl}(V)$. If f has this property at each point of X, then f is said to be weakly $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semicontinuous.

Remark 5.7

For a function $f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following implication holds: almost $({\tau }_1,{\tau }_2)\delta$-semicontinuity ⇒ weak $({\tau }_1,{\tau }_2)\delta$-semicontinuity.

The converse of the implication is not true in general. We give an example for the implication as follows.

Example 5.8

Let $X=\{a,b,c,d\}$ with topologies

$${\tau }_1=\{\varnothing ,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\},\{a,c,d\},X\}$$

and ${\tau }_2=\{\varnothing ,\{a\},\{c\},\{a,b\},\{a,c\},\{a,b,c\},\{a,c,d\},X\}$. Let

$$Y=\{-2,-1,0,1\}$$

with topologies

$${\sigma }_1=\{\varnothing ,\{-2\},\{0\},\{-2,-1\},\{-2,0\},\{-2,-1,0\},\{-2,0,1\},Y\}$$

and

$${\sigma }_2=\{\varnothing ,\{-2\},\{0\},\{-1\},\{-2,-1\},\{-2,0\},\{-1,0\},\{-2,-1,0\},\{-2,0,1\},Y\}.$$

Define a function $f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ as follows: $f(a)=-2$, $f(b)=0$, $f(c)=-1$ and $f(d)=1$. Then f is weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous but f is not almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

Corollary 5.9

For a function $f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)f is weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous at $x\in X$;
• (2)$x\in ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y containing $f(x)$;
• (3)$x\in {\tau }_1\mathit{\text{-Cl}}({\tau }_2{\mathit{\text{-Int}}}_{\delta }(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y containing $f(x)$.

Theorem 5.10

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)for each $x\in X$ and each ${\sigma }_1{\sigma }_2$-open subset V of Y such that $x\in F^{+}(V)$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $x\in F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$;
• (3)$F^{+}(V)\subseteq {\tau }_1\mathit{\text{-Cl}}({\tau }_2{\mathit{\text{-Int}}}_{\delta }(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (4)${\tau }_1\mathit{\text{-Int}}({\tau }_2{\mathit{\text{-Cl}}}_{\delta }(F^{-}(V)))\subseteq F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (5)${\tau }_1\mathit{\text{-Int}}({\tau }_2{\mathit{\text{-Cl}}}_{\delta }(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(K))))\subseteq F^{-}(K))$ for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (6)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(K)))\subseteq F^{-}(K)$ for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (7)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(B))))\subseteq F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(B))$ for every subset B of Y;
• (8)$F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(B))\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(B))))$ for every subset B of Y;
• (9)$F^{+}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (10)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}(V))\subseteq F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y.

Proof

$(1)\Rightarrow (2)$: Let $x\in X$ and V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $F(x)$. Since F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq {\sigma }_1{\sigma }_2\text{-Cl}(V)$ and hence $x\in F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V))$.

$(2)\Rightarrow (1)$: Let $x\in X$ and V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $F(x)$. By (2), there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $x\in F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V))$; hence $F(U)\subseteq {\sigma }_1{\sigma }_2\text{-Cl}(V)$. This shows that F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

$(1)\Rightarrow (3)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y and $x\in F^{+}(V)$. Then $F(x)\subseteq V$. By (1), there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq {\sigma }_1{\sigma }_2\text{-Cl}(V)$. Since U is $({\tau }_1,{\tau }_2)\delta$-semiopen, we have $x\in U\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(U))\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V))))$ and hence $F^{+}(V)\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V))))$.

$(3)\Rightarrow (4)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y. By (3), we have

$$X-F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(V))=F^{+}(Y-{\sigma }_1{\sigma }_2\text{-Cl}(V))\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(Y-{\sigma }_1{\sigma }_2\text{-Cl}(V)))))={\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(F^{+}(Y-{\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V)))))\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(F^{+}(Y-V)))={\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(X-F^{-}(V)))=X-{\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(F^{-}(V))).$$

Consequently, we obtain ${\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(F^{-}(V)))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(V))$.

$(4)\Rightarrow (5)$: Let K be any ${\sigma }_1{\sigma }_2$-closed subset of Y. Then ${\sigma }_1{\sigma }_2\text{-Int}(K)$ is ${\sigma }_1{\sigma }_2$-open in Y. By (4),

$${\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(F^{-}({\sigma }_1{\sigma }_2\text{-Int}(K))))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-}\text{Int}(K)))\subseteq F^{-}(K).$$

$(5)\Rightarrow (6)$: Let K be any ${\sigma }_1{\sigma }_2$-closed subset of Y. By (5), we have

$${\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(F^{-}({\sigma }_1{\sigma }_2\text{-Int}(K))))\subseteq F^{-}(K))$$

and $F^{-}({\sigma }_1{\sigma }_2\text{-Int}(K))\subseteq F^{-}(K)$. Since the equality

$$({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-Int}(K)))=F^{-}({\sigma }_1{\sigma }_2\text{-Int}(K))\cup {\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(F^{-}({\sigma }_1{\sigma }_2\text{-Int}(K))))$$

holds, we obtain $({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-Int}(K)))\subseteq F^{-}(K)$.

$(6)\Rightarrow (7)$: Let B be any subset of Y. Then ${\sigma }_1{\sigma }_2\text{-Cl}(B)$ is ${\sigma }_1{\sigma }_2$-closed in Y and by (6), $({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(B))))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(B))$.

$(7)\Rightarrow (8)$: Let B be any subset of Y. By (7), we have

$$F^{+}({\sigma }_1{\sigma }_2\text{-Int}(B))=X-F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(Y-B))\subseteq X-({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(Y-B))))=({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}(B)))).$$

$(8)\Rightarrow (9)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y. By (8),

$$F^{+}(V)=F^{+}({\sigma }_1{\sigma }_2\text{-Int}(V))\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}(V))))=({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V))).$$

$(9)\Rightarrow (10)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y. By (9), we get

$$({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}(V))\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))=({\tau }_1,{\tau }_2)\delta \text{-sCl}(X-F^{+}(Y-{\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))=X-({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(Y-{\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))=X-({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(Y-{\sigma }_1{\sigma }_2\text{-Cl}(V))))\subseteq X-F^{+}(Y-{\sigma }_1{\sigma }_2\text{-Cl}(V))=F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(V)).$$

Thus, $({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}(V))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(V))$.

$(10)\Rightarrow (1)$: Let $x\in X$ and V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $F(x)$. By (10), we have

$$({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V)))=X-({\tau }_1,{\tau }_2){\text{-sCl}}_{\delta }(F^{-}(Y-{\sigma }_1{\sigma }_2\text{-Cl}(V)))\supseteq X-F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(Y-{\sigma }_1{\sigma }_2\text{-Cl}(V)))=F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V)))\supseteq F^{+}(V).$$

Put $U=({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V)))$. Then U is a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq {\sigma }_1{\sigma }_2\text{-Cl}(V)$. This shows that F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous. □

Theorem 5.11

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is lower weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)for each $x\in X$ and each ${\sigma }_1{\sigma }_2$-open subset V of Y such that $x\in F^{-}(V)$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $x\in F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$;
• (3)$F^{-}(V)\subseteq {\tau }_1\mathit{\text{-Cl}}({\tau }_2{\mathit{\text{-Int}}}_{\delta }(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (4)${\tau }_1\mathit{\text{-Int}}({\tau }_2{\mathit{\text{-Cl}}}_{\delta }(F^{+}(V)))\subseteq F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (5)${\tau }_1\mathit{\text{-Int}}({\tau }_2{\mathit{\text{-Cl}}}_{\delta }(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(K))))\subseteq F^{+}(K))$ for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (6)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(K)))\subseteq F^{+}(K)$ for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (7)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(B))))\subseteq F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(B))$ for every subset B of Y;
• (8)$F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(B))\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(B))))$ for every subset B of Y;
• (9)$F^{-}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (10)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}(V))\subseteq F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y.

Proof

The proof is similar to that of Theorem 5.10. □

Corollary 5.12

For a function $f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)f is weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)for each $x\in X$ and each ${\sigma }_1{\sigma }_2$-open subset V of Y such that $x\in f^{-1}(V)$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $x\in f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$;
• (3)$f^{-1}(V)\subseteq {\tau }_1\mathit{\text{-Cl}}({\tau }_2{\mathit{\text{-Int}}}_{\delta }(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (4)${\tau }_1\mathit{\text{-Int}}({\tau }_2{\mathit{\text{-Cl}}}_{\delta }(f^{-1}(V)))\subseteq f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (5)${\tau }_1\mathit{\text{-Int}}({\tau }_2{\mathit{\text{-Cl}}}_{\delta }(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(K))))\subseteq f^{-1}(K))$ for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (6)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(K)))\subseteq f^{-1}(K)$ for every ${\sigma }_1{\sigma }_2$-closed subset K of Y;
• (7)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(B))))\subseteq f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(B))$ for every subset B of Y;
• (8)$f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(B))\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(B))))$ for every subset B of Y;
• (9)$f^{-1}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V)))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (10)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}(V))\subseteq f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y.

Definition 5.13

[33] Let A be a subset of a bitopological space $(X,{\tau }_1,{\tau }_2)$. A point $x\in X$ is called a $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\theta }$-cluster point of A if ${\tau }_1{\tau }_2\text{-Cl}(U)\cap A\ne \varnothing$ for every ${\tau }_1{\tau }_2$-open set U containing x. The set of all $({\tau }_1,{\tau }_2)\theta$-cluster point of A is called the $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\theta }$-closure of A and is denoted by $({\tau }_1,{\tau }_2)\theta \text{-Cl}(A)$.

A subset A of a bitopological space $(X,{\tau }_1,{\tau }_2)$ is said to be $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\theta }$-closed if $A=({\tau }_1,{\tau }_2)\theta \text{-Cl}(A)$. The complement of a $({\tau }_1,{\tau }_2)\theta$-closed set is said to be $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\theta }$-open. The union of all $({\tau }_1,{\tau }_2)\theta$-open sets contained in A is called the $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\theta }$-interior of A and is denoted by $({\tau }_1,{\tau }_2)\theta \text{-Int}(A)$.

Lemma 5.14

[33] For a subset A of a bitopological space $(X,{\tau }_1,{\tau }_2)$, the following properties hold:

• (1)If A is ${\tau }_1{\tau }_2$-open in X, then ${\tau }_1{\tau }_2\mathit{\text{-Cl}}(A)=({\tau }_1,{\tau }_2)\theta \mathit{\text{-Cl}}(A)$.
• (2)$({\tau }_1,{\tau }_2)\theta \mathit{\text{-Cl}}(A)$ is ${\tau }_1{\tau }_2$-closed in X.

Theorem 5.15

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(({\sigma }_1,{\sigma }_2)\theta \mathit{\text{-Cl}}(B))))\subseteq F^{-}(({\sigma }_1,{\sigma }_2)\theta \mathit{\text{-Cl}}(B))$ for every subset B of Y;
• (3)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(B))))\subseteq F^{-}(({\sigma }_1,{\sigma }_2)\theta \mathit{\text{-Cl}}(B))$ for every subset B of Y;
• (4)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))\subseteq F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (5)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))\subseteq F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y;
• (6)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(K)))\subseteq F^{-}(K)$ for every $({\sigma }_1,{\sigma }_2)r$-closed subset K of Y.

Proof

$(1)\Rightarrow (2)$: Let B be any subset of Y. By Lemma 5.14(2), $({\sigma }_1,{\sigma }_2)\theta \text{-Cl}(B)$ is ${\sigma }_1{\sigma }_2$-closed in Y. By Lemma 3.2(1) and Theorem 5.10, we obtain

$$\begin{gathered} ({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-Int}(({\sigma }_1,{\sigma }_2)\theta \text{-Cl}(B))))=F^{-}({\sigma }_1{\sigma }_2\text{-Int}(({\sigma }_1,{\sigma }_2)\theta \text{-Cl}(B)))\cup {\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(F^{-}({\sigma }_1{\sigma }_2\text{-Int}(({\sigma }_1,{\sigma }_2)\theta \text{-} \\ \text{Cl}(B)))))\subseteq F^{-}(({\sigma }_1,{\sigma }_2)\theta \text{-Cl}(B)). \end{gathered}$$

$(2)\Rightarrow (3)$: This is obvious since ${\sigma }_1{\sigma }_2\text{-Cl}(B)\subseteq ({\sigma }_1,{\sigma }_2)\theta \text{-Cl}(B)$ for every subset B of Y.

$(3)\Rightarrow (4)$: This is obvious since ${\sigma }_1{\sigma }_2\text{-Cl}(V)=({\sigma }_1,{\sigma }_2)\theta \text{-Cl}(V)$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y.

$(4)\Rightarrow (5)$: Let V be any $({\sigma }_1,{\sigma }_2)p$-open subset of Y. Then, we have $V\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$ and ${\sigma }_1{\sigma }_2\text{-Cl}(V)={\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V)))$. Now, put $G={\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$, then G is ${\sigma }_1{\sigma }_2$-open in Y and ${\sigma }_1{\sigma }_2\text{-Cl}(G)={\sigma }_1{\sigma }_2\text{-Cl}(V)$. By (4), $({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(V))$.

$(5)\Rightarrow (6)$: Let K be any $({\sigma }_1,{\sigma }_2)r$-closed subset of Y. Then, we have ${\sigma }_1{\sigma }_2\text{-Int}(K)$ is $({\sigma }_1,{\sigma }_2)p$-open in Y and by (5),

$$({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-Int}(K)))=({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}(K)))))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}(K)))=F^{-}(K).$$

$(6)\Rightarrow (1)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y. Then ${\sigma }_1{\sigma }_2\text{-Cl}(V)$ is $({\sigma }_1,{\sigma }_2)r$-closed in Y and by (6),

$$({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}(V))\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(V)).$$

It follows from Theorem 5.10 that F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous. □

Theorem 5.16

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is lower weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(({\sigma }_1,{\sigma }_2)\theta \mathit{\text{-Cl}}(B))))\subseteq F^{+}(({\sigma }_1,{\sigma }_2)\theta \mathit{\text{-Cl}}(B))$ for every subset B of Y;
• (3)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(B))))\subseteq F^{+}(({\sigma }_1,{\sigma }_2)\theta \mathit{\text{-Cl}}(B))$ for every subset B of Y;
• (4)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))\subseteq F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (5)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))\subseteq F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y;
• (6)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(K)))\subseteq F^{+}(K)$ for every $({\sigma }_1,{\sigma }_2)r$-closed subset K of Y.

Proof

The proof is similar to that of Theorem 5.15. □

Corollary 5.17

For a function $f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)f is weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(({\sigma }_1,{\sigma }_2)\theta \mathit{\text{-Cl}}(B))))\subseteq f^{-1}(({\sigma }_1,{\sigma }_2)\theta \mathit{\text{-Cl}}(B))$ for every subset B of Y;
• (3)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(B))))\subseteq f^{-1}(({\sigma }_1,{\sigma }_2)\theta \mathit{\text{-Cl}}(B))$ for every subset B of Y;
• (4)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))\subseteq f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every ${\sigma }_1{\sigma }_2$-open subset V of Y;
• (5)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))\subseteq f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y;
• (6)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}(K)))\subseteq f^{-1}(K)$ for every $({\sigma }_1,{\sigma }_2)r$-closed subset K of Y.

Theorem 5.18

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))\subseteq F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)\beta$-open subset V of Y;
• (3)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))\subseteq F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)s$-open subset V of Y;
• (4)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))\subseteq F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y.

Proof

$(1)\Rightarrow (2)$: Let V be any $({\sigma }_1,{\sigma }_2)\beta$-open subset of Y. Then

$$V\subseteq {\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V)))$$

and ${\sigma }_1{\sigma }_2\text{-Cl}(V)={\sigma }_1{\sigma }_2\text{-Cl}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V)))$. Since ${\sigma }_1{\sigma }_2\text{-Cl}(V)$ is a $({\sigma }_1,{\sigma }_2)r$-closed set, by Theorem 5.10 we have

$$({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-}\text{Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(V)).$$

$(2)\Rightarrow (3)$: Let V be any $({\sigma }_1,{\sigma }_2)s$-open subset of Y. Then V is $({\sigma }_1,{\sigma }_2)\beta$-open in Y. By (2), we have $({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(V))$.

$(3)\Rightarrow (4)$: Let V be any $({\sigma }_1,{\sigma }_2)p$-open subset of Y. Then ${\sigma }_1{\sigma }_2\text{-Cl}(V)$ is $({\sigma }_1,{\sigma }_2)r$-closed and ${\sigma }_1{\sigma }_2\text{-Cl}(V)$ is $({\sigma }_1,{\sigma }_2)s$-open. By (3),

$$({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(V)).$$

$(4)\Rightarrow (1)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y. Then V is $({\sigma }_1,{\sigma }_2)p$-open in Y and by (4), we have

$$({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}(V))\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(V)).$$

It follows from Theorem 5.10 that F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous. □

Theorem 5.19

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is lower weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))\subseteq F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)\beta$-open subset V of Y;
• (3)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))\subseteq F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)s$-open subset V of Y;
• (4)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))\subseteq F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y.

Proof

The proof is similar to that of Theorem 5.18. □

Corollary 5.20

For a function $f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)f is weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))\subseteq f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)\beta$-open subset V of Y;
• (3)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))\subseteq f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)s$-open subset V of Y;
• (4)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Int}}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))))\subseteq f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y.

Theorem 5.21

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)${\tau }_1\mathit{\text{-Int}}({\tau }_2{\mathit{\text{-Cl}}}_{\delta }(F^{-}(V)))\subseteq F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y;
• (3)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{-}(V))\subseteq F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y;
• (4)$F^{+}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V)))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y.

Proof

$(1)\Rightarrow (2)$: Let V be any $({\sigma }_1,{\sigma }_2)p$-open subset of Y. Since F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous, by Theorem 5.10 we obtain

$${\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(F^{-}(V)))\subseteq {\tau }_1\text{-Cl}({\tau }_2{\text{-Int}}_{\delta }(F^{-}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V)))))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(V)).$$

$(2)\Rightarrow (3)$: Let V be any $({\sigma }_1,{\sigma }_2)p$-open subset of Y. By (2) and Lemma 3.2(1), $({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}(V))={\tau }_1\text{-Int}({\tau }_2{\text{-Cl}}_{\delta }(F^{-}(V)))\cup F^{-}(V)\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(V))$.

$(3)\Rightarrow (4)$: Let V be any $({\sigma }_1,{\sigma }_2)p$-open subset of Y. By (3), we have

$$X-({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V)))=({\tau }_1,{\tau }_2)\delta \text{-sCl}(X-F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V)))=({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{-}(Y-{\sigma }_1{\sigma }_2\text{-Cl}(V)))\subseteq F^{-}({\sigma }_1{\sigma }_2\text{-Cl}(Y-{\sigma }_1{\sigma }_2\text{-Cl}(V)))=X-F^{+}({\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V)))\subseteq X-F^{+}(V)$$

and hence $F^{+}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V)))$.

$(4)\Rightarrow (1)$: Let V be any ${\sigma }_1{\sigma }_2$-open subset of Y. Then V is $({\sigma }_1,{\sigma }_2)p$-open in Y and by (4), $F^{+}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V)))$. Therefore, by Theorem 5.10, F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous. □

Theorem 5.22

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)F is lower weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)${\tau }_1\mathit{\text{-Int}}({\tau }_2{\mathit{\text{-Cl}}}_{\delta }(F^{+}(V)))\subseteq F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y;
• (3)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(F^{+}(V))\subseteq F^{+}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y;
• (4)$F^{-}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(F^{-}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V)))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y.

Proof

The proof is similar to that of Theorem 5.21. □

Corollary 5.23

For a function $f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties are equivalent:

• (1)f is weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)${\tau }_1\mathit{\text{-Int}}({\tau }_2{\mathit{\text{-Cl}}}_{\delta }(f^{-1}(V)))\subseteq f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y;
• (3)$({\tau }_1,{\tau }_2)\delta \mathit{\text{-sCl}}(f^{-1}(V))\subseteq f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y;
• (4)$f^{-1}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \mathit{\text{-sInt}}(f^{-1}({\sigma }_1{\sigma }_2\mathit{\text{-Cl}}(V)))$ for every $({\sigma }_1,{\sigma }_2)p$-open subset V of Y.

Definition 5.24

A bitopological space $(X,{\tau }_1,{\tau }_2)$ is said to be:

• (1)almost-$({\mathit{\tau }}_1,{\mathit{\tau }}_2)$-regular if for each $({\tau }_1,{\tau }_2)r$-closed subset F of X and each $x\in X-F$, there exist disjoint ${\tau }_1{\tau }_2$-open sets U and V such that $x\in U$ and $F\subseteq V$;
• (2)semi-$({\mathit{\tau }}_1,{\mathit{\tau }}_2)$-regular if for each ${\tau }_1{\tau }_2$-open subset U of X and each $x\in U$, there exists a $({\tau }_1,{\tau }_2)r$-open subset V of X such that $x\in V\subseteq U$.

Lemma 5.25

If $(X,{\tau }_1,{\tau }_2)$ is almost-$({\tau }_1,{\tau }_2)$-regular, then for each $x\in X$ and each $({\tau }_1,{\tau }_2)r$-open subset V of X containing x, there exists a $({\tau }_1,{\tau }_2)r$-open subset U of X such that $x\in U\subseteq {\tau }_1{\tau }_2\mathit{\text{-Cl}}(U)\subseteq V$.

Proof

Let V be any $({\tau }_1,{\tau }_2)r$-open subset of X containing x. Then, $x\notin X-V$. Therefore, there exist disjoint ${\tau }_1{\tau }_2$-open sets $U_1$ and $U_2$ such that $x\in U_1$ and $X-V\subseteq U_2$. Then ${\tau }_1{\tau }_2\text{-Cl}(U_1)\cap U_2=\varnothing$ and hence ${\tau }_1{\tau }_2\text{-Cl}(U_1)\subseteq X-U_2\subseteq V$. Thus, $x\in U_1\subseteq {\tau }_1{\tau }_2\text{-Cl}(U_1)\subseteq V$. Again, $U_1\subseteq {\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}(U_1))\subseteq {\tau }_1{\tau }_2\text{-Cl}(U_1)\subseteq V$. Therefore, if ${\tau }_1{\tau }_2\text{-Int}({\tau }_1{\tau }_2\text{-Cl}(U_1))=U$, then U is $({\tau }_1,{\tau }_2)r$-open such that $U_1\subseteq U\subseteq {\tau }_1{\tau }_2\text{-Cl}(U)={\tau }_1{\tau }_2\text{-Cl}(U_1)\subseteq V$. This shows that $x\in U\subseteq {\tau }_1{\tau }_2\text{-Cl}(U)\subseteq V$. □

Theorem 5.26

For a function $f:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$, the following properties hold:

• (1)If f is weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous and $(Y,{\sigma }_1,{\sigma }_2)$ is almost-$({\sigma }_1,{\sigma }_2)$-regular, then f is almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous.
• (2)If f is almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous and $(Y,{\sigma }_1,{\sigma }_2)$ is semi-$({\sigma }_1,{\sigma }_2)$-regular, then f is $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

Proof

(1) Suppose that f is weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous and $(Y,{\sigma }_1,{\sigma }_2)$ is almost-$({\sigma }_1,{\sigma }_2)$-regular. Let $x\in X$ and V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $f(x)$. By Lemma 5.25, there exists a $({\sigma }_1,{\sigma }_2)r$-open subset U of Y such that $f(x)\subseteq U\subseteq {\sigma }_1{\sigma }_2\text{-Cl}(U)\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$. Since f is weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous, there exists a $({\tau }_1,{\tau }_2)\delta$-open subset G of X containing x such that $f(G)\subseteq {\sigma }_1{\sigma }_2\text{-Cl}(U)\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V))$. This shows that f is almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

(2) Suppose that f is almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous and $(Y,{\sigma }_1,{\sigma }_2)$ is semi-$({\sigma }_1,{\sigma }_2)$-regular. Let $x\in X$ and V be any ${\sigma }_1{\sigma }_2$-open subset of Y containing $f(x)$. By the semi-$({\sigma }_1,{\sigma }_2)$-regularity of Y, there exists a $({\sigma }_1,{\sigma }_2)r$-open subset U of Y such that $f(x)\in U\subseteq V$. Since f is almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset G of X containing x such that $f(G)\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(U))=U\subseteq V$. This shows that f is $({\tau }_1,{\tau }_2)\delta$-semicontinuous. □

Definition 5.27

A bitopological space $(X,{\tau }_1,{\tau }_2)$ is said to be ${\mathit{\tau }}_1{\mathit{\tau }}_2$-connected [2] (resp. $({\tau }_1,{\tau }_2)\delta$-semiconnected) if X cannot be written as the union of two non-empty disjoint ${\tau }_1{\tau }_1$-open (resp. $({\tau }_1,{\tau }_2)\delta$-semiopen) sets.

Definition 5.28

[2] A subset A of a bitopological space $(X,{\tau }_1,{\tau }_2)$ is called ${\mathit{\tau }}_1{\mathit{\tau }}_2$-clopen if A is both ${\tau }_1{\tau }_2$-open and ${\tau }_1{\tau }_2$-closed.

Theorem 5.29

If $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ is an upper or lower weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous surjective multifunction such that $F(x)$ is ${\sigma }_1{\sigma }_2$-connected for each $x\in X$ and $(X,{\tau }_1,{\tau }_2)$ is $({\tau }_1,{\tau }_2)\delta$-semiconnected, then $(Y,{\sigma }_1,{\sigma }_2)$ is ${\sigma }_1{\sigma }_2$-connected.

Proof

Suppose that $(Y,{\sigma }_1,{\sigma }_2)$ is not ${\sigma }_1{\sigma }_2$-connected. There exist non-empty ${\sigma }_1{\sigma }_2$-open subsets U and V of Y such that $U\cap V=\varnothing$ and

$$U\cup V=Y.$$

Since $F(x)$ is ${\sigma }_1{\sigma }_2$-connected for each $x\in X$, either $F(x)\subseteq U$ or $F(x)\subseteq V$. If $x\in F^{+}(U\cup V)$, then $F(x)\subseteq U\cup V$ and hence $x\in F^{+}(U)\cup F^{+}(V)$. Moreover, since F is surjective, there exist x and y in X such that $F(x)\subseteq U$ and $F(y)\subseteq V$; hence $x\in F^{+}(U)$ and $y\in F^{+}(V)$. Therefore, we obtain the following:

• (1)$F^{+}(U)\cup F^{+}(V)=F^{+}(U\cup V)=X$;
• (2)$F^{+}(U)\cap F^{+}(V)=F^{+}(U\cap V)=\varnothing$;
• (3)$F^{+}(U)\ne \varnothing$ and $F^{+}(V)\ne \varnothing$.

Next, we show that $F^{+}(U)$ and $F^{+}(V)$ are $({\tau }_1,{\tau }_2)\delta$-semiopen in X.

(i) Let F be upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous. By Theorem 5.10, we obtain

$$F^{+}(V)\subseteq ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V)))=({\tau }_1,{\tau }_2)\delta \text{-}\text{sInt}(F^{+}(V))$$

since V is ${\sigma }_1{\sigma }_2$-clopen. Therefore,

$$F^{+}(V)=({\tau }_1,{\tau }_2)\delta \text{-}\text{sInt}(F^{+}(V))$$

and hence $F^{+}(V)$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X. Similarly, we obtain $F^{+}(U)$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X. This shows that $(X,{\tau }_1,{\tau }_2)$ is not $({\tau }_1,{\tau }_2)\delta$-semiconnected.

$(ii)$ Let F be lower weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous. By Theorem 5.11, we obtain $({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{+}(V))\subseteq F^{+}({\sigma }_1{\sigma }_2\text{-Cl}(V))=F^{+}(V)$ since V is ${\sigma }_1{\sigma }_2$-clopen. Therefore, $F^{+}(V)=({\tau }_1,{\tau }_2)\delta \text{-sCl}(F^{+}(V))$ and hence $F^{+}(V)$ is $({\tau }_1,{\tau }_2)\delta$-semiclosed in X. Thus, we have $F^{+}(U)$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X. Similarly, we obtain $F^{+}(V)$ is $({\tau }_1,{\tau }_2)\delta$-semiopen in X. Therefore, $(X,{\tau }_1,{\tau }_2)$ is not $({\tau }_1,{\tau }_2)\delta$-semiconnected. This completes the proof. □

Definition 5.30

Let A be a subset of a bitopological space $(X,{\tau }_1,{\tau }_2)$. The $({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semifrontior of a subset A, denoted by $({\tau }_1,{\tau }_2)\delta \text{-sFr}(A)$, is defined by

$$({\tau }_1,{\tau }_2)\delta \text{-sFr}(A)=({\tau }_1,{\tau }_2)\delta \text{-sCl}(A)\cap ({\tau }_1,{\tau }_2)\delta \text{-sCl}(X-A)=({\tau }_1,{\tau }_2)\delta \text{-sCl}(A)-({\tau }_1,{\tau }_2)\delta \text{-sInt}(A).$$

Theorem 5.31

The set of all points of X at which a multifunction

$$F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\tau }_2)$$

is not upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous is identical with the union of the $({\tau }_1,{\tau }_2)\delta$-semifrontier of the upper inverse images of the ${\sigma }_1{\sigma }_2$-closure of ${\sigma }_1{\sigma }_2$-open sets containing $F(x)$.

Proof

Let $x\in X$ at which F is not upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous. Then, there exists a ${\sigma }_1{\sigma }_2$-open subset V of Y containing $F(x)$ such that $U\cap (X-F^{+}(V))\ne \varnothing$ for every $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x. Therefore, we have

$$x\in ({\tau }_1,{\tau }_2)\delta \text{-sCl}(X-F^{+}(V))=X-({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(V))$$

and $x\in F^{+}(V)$. Thus, we obtain $x\in ({\tau }_1,{\tau }_2)\delta \text{-sFr}(F^{+}(V))$.

Conversely, suppose that V is a ${\sigma }_1{\sigma }_2$-open subset of Y containing $F(x)$ such that $x\in ({\tau }_1,{\tau }_2)\delta \text{-sFr}(F^{+}(V))$. If F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous at x, then there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $U\subseteq F^{+}(V)$; hence $x\in ({\tau }_1,{\tau }_2)\delta \text{-sInt}(F^{+}(V))$. This is a contradiction and hence F is not upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous at x. □

Theorem 5.32

The set of all points of X at which a multifunction

$$F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\tau }_2)$$

is not lower weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous is identical with the union of the $({\tau }_1,{\tau }_2)\delta$-semifrontier of the lower inverse images of the ${\sigma }_1{\sigma }_2$-closure of ${\sigma }_1{\sigma }_2$-open sets meeting $F(x)$.

Proof

The proof is similar to that of Theorem 5.31. □

Definition 5.33

A multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\tau }_2)$ is called punctually $({\mathit{\tau }}_1,{\mathit{\tau }}_2)$-closed if $F(x)$ is ${\sigma }_1{\sigma }_2$-closed in Y for each point $x\in X$.

Definition 5.34

A bitopological space $(X,{\tau }_1,{\tau }_2)$ is said to be:

• (1)$({\mathit{\tau }}_1,{\mathit{\tau }}_2)\mathit{\delta }$-semi-$T_2$ if for each distinct points $x,y\in X$, there exist $({\tau }_1,{\tau }_2)\delta$-semiopen subset U and V of X containing x and y, respectively, such that $U\cap V=\varnothing$;
• (2)${\mathit{\tau }}_1{\mathit{\tau }}_2$-normal if for any pair of disjoint ${\tau }_1{\tau }_2$-closed subsets F and K of X, there exist disjoint ${\tau }_1{\tau }_2$-open sets U and V such that $F\subseteq U$ and $K\subseteq V$.

Theorem 5.35

Let $(X,{\tau }_1,{\tau }_2)$ be a bitopological space. If for each pair of distinct points $x_1$ and $x_2$ in X, there exists a multifunction

$$F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2),$$

where $(Y,{\sigma }_1,{\sigma }_2)$ is a ${\sigma }_1{\sigma }_2$-normal space, such that

• (1)F is punctually $({\sigma }_1,{\sigma }_2)$-closed,
• (2)F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous at $x_1$,
• (3)F is upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous at $x_2$, and
• (4)$F(x_1)\cap F(x_2)=\varnothing$,

then $(X,{\tau }_1,{\tau }_2)$ is a $({\tau }_1,{\tau }_2)\delta$-semi-$T_2$ space.

Proof

Let $x_1$ and $x_2$ be distinct points of X. Then, since $(Y,{\sigma }_1,{\sigma }_2)$ is a ${\sigma }_1{\sigma }_2$-normal space, F is punctually $({\sigma }_1,{\sigma }_2)$-closed and

$$F(x_1)\cap F(x_2)=\varnothing ,$$

there exist ${\sigma }_1{\sigma }_2$-open subsets $V_1$ and $V_2$ of Y containing $F(x_1)$ and $F(x_2)$, respectively, such that $V_1\cap V_2=\varnothing$; hence

$${\sigma }_1{\sigma }_2\text{-Cl}(V_1)\cap {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V_2))=\varnothing .$$

Since F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous at $x_1$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset $U_1$ of X containing $x_1$ such that $F(U_1)\subseteq {\sigma }_1{\sigma }_2\text{-Cl}(V_1)$. Since F is upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous at $x_2$, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset $U_2$ of X containing $x_2$ such that $F(U_2)\subseteq {\sigma }_1{\sigma }_2\text{-Int}({\sigma }_1{\sigma }_2\text{-Cl}(V_2))$. Therefore, we have $F(U_1)\cap F(U_2)=\varnothing$ which implies $U_1\cap U_2=\varnothing$. This shows that $(X,{\tau }_1,{\tau }_2)$ is a $({\tau }_1,{\tau }_2)\delta$-semi-$T_2$ space. □

Definition 5.36

[2] A collection $\mathcal{U}$ of subsets of a bitopological space $(X,{\tau }_1,{\tau }_2)$ is said to be ${\mathit{\tau }}_1{\mathit{\tau }}_2$-locally finite if every $x\in X$ has a ${\tau }_1{\tau }_2$-neighborhood which intersects only finitely many elements of $\mathcal{U}$.

Definition 5.37

[2] A subset A of a bitopological space $(X,{\tau }_1,{\tau }_2)$ is said to be:

• (1)${\mathit{\tau }}_1{\mathit{\tau }}_2$-paracompact if every cover of A by ${\tau }_1{\tau }_2$-open sets of X is refined by a cover of A which consists of ${\tau }_1{\tau }_2$-open sets of X and is ${\tau }_1{\tau }_2$-locally finite in X;
• (2)${\mathit{\tau }}_1{\mathit{\tau }}_2$-regular if for each $x\in A$ and each ${\tau }_1{\tau }_2$-open set U of X containing x, there exists a ${\tau }_1{\tau }_2$-open set V of X such that $x\in V\subseteq {\tau }_1{\tau }_2\text{-Cl}(V)\subseteq U$.

Lemma 5.38

[2] If A is a ${\tau }_1{\tau }_2$-regular ${\tau }_1{\tau }_2$-paracompact set of a bitopological space $(X,{\tau }_1,{\tau }_2)$ and U is a ${\tau }_1{\tau }_2$-open neighborhood of A, then there exists a ${\tau }_1{\tau }_2$-open set V of X such that $A\subseteq V\subseteq {\tau }_1{\tau }_2\mathit{\text{-Cl}}(V)\subseteq U$.

Theorem 5.39

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ such that $F(x)$ is ${\sigma }_1{\sigma }_2$-regular ${\sigma }_1{\sigma }_2$-paracompact for each $x\in X$, the following properties are equivalent:

• (1)F is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)F is upper almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (3)F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

Proof

We prove only the implication $(3)\Rightarrow (1)$ since the others are obvious. Suppose that F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous. Let $x\in X$ and V be any ${\sigma }_1{\sigma }_2$-open subset of Y such that $F(x)\subseteq V$. Since $F(x)$ is ${\sigma }_1{\sigma }_2$-regular ${\sigma }_1{\sigma }_2$-paracompact, by Lemma 5.38 there exists a ${\sigma }_1{\sigma }_2$-open subset G of Y such that $F(x)\subseteq G\subseteq {\sigma }_1{\sigma }_2\text{-Cl}(G)\subseteq V$. Since F is upper weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous at x, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that $F(U)\subseteq {\sigma }_1{\sigma }_2\text{-Cl}(G)$ and hence $F(U)\subseteq V$. This shows that F is upper $({\tau }_1,{\tau }_2)\delta$-semicontinuous. □

Lemma 5.40

[33] Let A be a subset of a bitopological space $(X,{\tau }_1,{\tau }_2)$. If A is ${\tau }_1{\tau }_2$-regular, then for every ${\tau }_1{\tau }_2$-open set G which intersects A, there exists a ${\tau }_1{\tau }_2$-open set W such that $A\cap W\ne \varnothing$ and ${\tau }_1{\tau }_2\mathit{\text{-Cl}}(W)\subseteq G$.

Theorem 5.41

For a multifunction $F:(X,{\tau }_1,{\tau }_2)\to (Y,{\sigma }_1,{\sigma }_2)$ such that $F(x)$ is ${\sigma }_1{\sigma }_2$-regular for each $x\in X$, the following properties are equivalent:

• (1)F is lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (2)F is lower almost $({\tau }_1,{\tau }_2)\delta$-semicontinuous;
• (3)F is lower weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous.

Proof

We prove only the implication $(3)\Rightarrow (1)$ since the others are obvious. Suppose that F is lower weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous. Let $x\in X$ and V be any ${\sigma }_1{\sigma }_2$-open subset of Y such that $F(x)\cap V\ne \varnothing$. Since $F(x)$ is ${\sigma }_1{\sigma }_2$-regular, by Lemma 5.40 there exists a ${\sigma }_1{\sigma }_2$-open subset G of Y such that $F(x)\cap G\ne \varnothing$ and ${\sigma }_1{\sigma }_2\text{-Cl}(G)\subseteq V$. Since F is lower weakly $({\tau }_1,{\tau }_2)\delta$-semicontinuous at x, there exists a $({\tau }_1,{\tau }_2)\delta$-semiopen subset U of X containing x such that ${\sigma }_1{\sigma }_2\text{-Cl}(G)\cap F(z)\ne \varnothing$ for every $z\in U$ and hence $V\cap F(z)\ne \varnothing$ for every $z\in U$ as well, and this shows that F is lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous at x. This shows that F is lower $({\tau }_1,{\tau }_2)\delta$-semicontinuous. □

Declarations

Author contribution statement

C. Boonpok: Conceived and designed the analysis; Analyzed and interpreted the data; Contributed analysis tools or data; Wrote the paper.

Funding statement

This research project was financially supported by Mahasarakham University.

Declaration of interests statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.