On C_ℵ-Fibrations in Bitopological Semigroups

Abstract

We extend the path lifting property in homotopy theory for topological spaces to bitopological semigroups and we show and prove its role in the C _ℵ-fibration property. We give and prove the relationship between the C _ℵ-fibration property and an approximate fibration property. Furthermore, we study the pullback maps for C _ℵ-fibrations.

1. Introduction

In homotopy theory for topological space (i.e., spaces), Hurewicz [1] introduced the concepts of fibrations and path lifting property of maps and showed its equivalence with the covering homotopy property. Coram and Duvall [2] introduced approximate fibrations as a generalization of cell-like maps [3] and showed that the uniform limit of a sequence of Hurewicz fibrations is an approximate fibration. In 1963, Kelly [4] introduced the notion of bitopological spaces. Such spaces were equipped with its two (arbitrary) topologies. The reader is suggested to refer to [4] for the detail definitions and notations. The concept of homotopy theory for topological semigroups has been introduced by Cerin in 2002 [5]. In this theory, he introduced S _ℵ-fibrations as extension of Hurewicz fibrations. In [6], we introduced the concepts of bitopological semigroups, c-bitopological semigroups, and C _ℵ-fibrations as extension of S _ℵ-fibrations.

This paper is organized as follows. It consists of five sections. After this Introduction, Section 2 is devoted to some preliminaries. In Section 3 we show the pullbacks of S-maps which have the C _ℵ-fibration property that will also have this property and the pullbacks of C _ℵ-fibrations are C _ℵ-fibrations under given conditions. In Section 4 we develop and extend path lifting property in homotopy theory for topological semigroups to theory for bitopological semigroups. Some results about Hurewicz fibrations carry over. In Section 5 we give and prove the relationship between the C _Nπ-fibration property and an approximate fibration property.

2. Preliminaries

Throughout this paper, by all X _τ we mean all topological spaces (X, τ) which will be assumed Hausdorff spaces. By all X _τ12 we mean all bitopological spaces (X, τ ₁, τ ₂). For two bitopological spaces X _τ12 and Y _ρ12, a p-map h : X _τ12 → Y _ρ12 is a function from X into Y that is continuous function (i.e., a map) from a space X _τ1 into a space Y _ρ1 and from X _τ2 into Y _ρ2 [4].

Recall [5] that a topological semigroup or an S-space is a pair (X _τ, ∗) consisting of a topological space X _τ and a map ∗ : X _τ × X _τ → X _τ from the product space X _τ × X _τ into X _τ such that ∗(x, ∗(y, z)) = ∗(∗(x, y), z) for all x, y, z ∈ X. An S-space (A, ∗′) is called an S-subspace of (X _τ, ∗) if A is a subspace of X _τ and the map ∗ takes the product A × A into A and ∗′(x, y) = ∗(x, y) for all x, y ∈ A. We denote the class of all S-spaces by ℵ. For every space X _τ, by P(X _τ), we mean the space of all paths from the unit closed interval I = [0,1] into X _τ with the compact-open topology. Recall [5] that, for every S-space (X _τ, ∗), (P(X _τ), p(∗)) is an S-space where p(∗) : P(X _τ) × P(X _τ) → P(X _τ) is a map defined by p(∗)(α, β)(t) = ∗(α(t), β(t)) for all α, β ∈ P(X _τ), t ∈ I. The shorter notion for this S-space will be P(X _τ, ∗). For every space X _τ, the natural S-space is an S-space (X _τ, π _i), where π _i is a continuous associative multiplication on X _τ given by π ₁(x, y) = x and π ₂(x, y) = y for all x, y ∈ X. We denote the class of all natural S-spaces (X _τ, π) by N _π, where π = π ₁, π ₂.

Recall [5] that the function f : (X _τ, ∗)→(O _ρ, ∘) is called an S-map if f is a map of a space X _τ into O _ρ and f(∗(x, y)) = ∘(f(x), f(y)) for all x, y ∈ X. The function f : X _τ → O _ρ of a natural S-space (X _τ, π) into (O _ρ, π) is an S-map if and only if it is continuous. The S-maps f, g : (X _τ, ∗)→(O _ρ, ∘) are called S-homotopic and write f ≃_s  g provided there is an S-map H : (X _τ, ∗) → P(O _ρ, ∘) called an S-homotopy such that H(x)(0) = f(x) and H(x)(1) = g(x) for all x ∈ X.

A bitopological semigroup is a pair (X _τ12, ∗) consisting of a bitopological space X _τ12 and the associative multiplication ∗ on X such that ∗ is an p-map from the product bitopological space (X × X, τ ₁ × τ ₁, τ ₂ × τ ₂) into X _τ12. For B⊆X, by B|_τ12 we mean the bitopological subspace (B, τ ₁|_B, τ ₂|_B) of X _τ12. If the p-map ∗ takes the product B × B into B then the pair (B|_τ12, ∗) will be a bitopological semigroup and will be called an b-subspace of (X _τ12, ∗).

The function h : (X _τ12, ∗)→(Y _ρ12, ∘) is called an S _i-map from (X _τ12, ∗) into (Y _ρ12, ∘) provided h is an S-map from a function S-space (X _τi, ∗) into an S-space (Y _ρi, ∘), where i = 1,2. We say that h is an Sp-map if it is an S ₁-map and S ₂-map.

An c-bitopological semigroup is a triple (X _τ12, ∗, X) consisting of bitopological semigroups (X _τ12, ∗) and an S-map X : (X _τ2, ∗)→(X _τ1, ∗) from an S-space (X _τ2, ∗) into an S-space (X _τ1, ∗). In our work, for any S-space, (O _ρ, ∘) can be regarded as an c-bitopological semigroup (O _ρρ, ∘, id) where id is the identity S-map on (O _ρ, ∘). That is, (O _ρ, ∘): = (O _ρρ, ∘, id).

An c-map from (X _τ12, ∗, X) into (O _ρ, ∘) is a pair f ₁₂ = (f ₁, f ₂):(X _τ12, ∗, X)→(O _ρ, ∘) of an S ₁-map f ₁ : (X _τ1, ∗)→(O _ρ, ∘) and S ₂-map f ₂ : (X _τ2, ∗)→(O _ρ, ∘) such that f ₁∘X = f ₂.

Definition 1 (see [6]). —

Let f : (X _τ, ∗)→(O _ρ, ∘) and h : (X _τ′′, ∗′)→(X _τ, ∗) be two S-maps. An S-map f is said to have the C _ℵ-fibration property by an S-map h provided for every (Y _ω, ⋆) ∈ ℵ and, given two S-maps g : (Y _ω, ⋆)→(X _τ′′, ∗′) and G : (Y _ω, ⋆) → P(O _ρ, ∘) with G ₀ = f∘(h∘g), there exists an S-homotopy H : (Y _ω, ⋆) → P(X _τ, ∗) such that H ₀ = h∘g and f∘H _t = G _t for all t ∈ I.

Definition 2 (see [6]). —

An c-map f ₁₂ : (X _τ12, ∗, X)→(O _ρ, ∘) is called an C _ℵ-fibration if an S ₁-map f ₁ : (X _τ1, ∗)→(O _ρ, ∘) has the C _ℵ-fibration property by an S-map X : (X _τ2, ∗)→(X _τ1, ∗). That is, for every (Y _ω, ⋆) ∈ ℵ and given two S-maps g : (Y _ω, ⋆)→(X _τ2, ∗) and G : (Y _ω, ⋆) → P(O _ρ, ∘) with G ₀ = f ₂∘g, there exists an S-homotopy H : (Y _ω, ⋆) → P(X _τ1, ∗) such that H ₀ = X∘g and f ₁∘H _t = G _t for all t ∈ I.

Let (X _τ12, ∗, B) be an c-bitopological semigroup and let (B|_τ12, ∗) be an b-subspace of (X _τ12, ∗). The c-bitopological semigroup (B|_τ12, ∗, B) is called an c-subspace of (X _τ12, ∗, X) provided B(b) = X(b) for all b ∈ B.

Theorem 3 (see [6]). —

Let f ₁₂ : (X _τ12, ∗, X)→(O _ρ, ∘) be an c-map and (B|_ρ, ∘) be an S-subspace of (O _ρ, ∘) such that f ₁ ⁻¹(B) = f ₂ ⁻¹(B). Then the triple (B ⁻|_τ12, ∗, X|_B⁻) is an c-subspace of (X _τ12, ∗, X) and a pair f ₁₂|_B⁻ = (f ₁|_B⁻, f ₂|_B⁻) is an c-map from an c-bitopological semigroup (B ⁻|_τ12, ∗, X|_B⁻) into (B|_ρ, ∘), where B ⁻ = f ₁ ⁻¹(B).

Corollary 4 (see [6]). —

Let f ₁₂ : (X _τ12, ∗, X)→(O _ρ, ∘) be an C _ℵ-fibration and let (B|_ρ, ∘) be an S-subspace of (O _ρ, ∘) such that f ₁ ⁻¹(B) = f ₂ ⁻¹(B). Then the restriction c-map

$$\begin{array}{c}{f_{\mathrm{12}}|}_{B^{-}}:({B^{-}|}_{{\tau }_{\mathrm{12}}},\ast ,{\mathcal{X}|}_{B^{-}})⟶\left({B|}_{\rho },\circ \right)\end{array}$$ (1)

is an C _ℵ-fibration, where B ⁻ = f ₁ ⁻¹(B).

3. The Pullback c-Maps

In this section, we show that the pullbacks of S-maps which have the C _ℵ-fibration property will also have this property and the pullbacks of C _ℵ-fibrations are C _ℵ-fibrations under given conditions.

Let f ₁₂ : (X _τ12, ∗, X)→(O _ρ, ∘) be an c-map and let h : (Q _υ, ⊙)→(O _ρ, ∘) be an S-map. Let

$$\begin{array}{c}{X}^{hi}=\{(x,b)\in X\times Q\hspace{0.17em}\mid \hspace{0.17em}{f}_i\left(x\right)=h\left(b\right)\},\end{array}$$ (2)

where i = 1,2.

Lemma 5 . —

Let f ₁₂ : (X _τ12, ∗, X)→(O _ρ, ∘) be an c-map and let h : (Q _υ, ⊙)→(O _ρ, ⊙) be an S-map. Then the pair (X ^hi|_τ12×υ, ∗ × ⊙) is an b-subspace of the bitopological semigroup ((X × Q)_τ12×υ, ∗ × ⊙), where i = 1,2.

Proof —

It is clear that X ^h1|_τ12×υ and X ^h2|_τ12×υ are subspaces of a bitopological space (X × O)_τ12×υ. Since h is an S-map and f ₁ is an S ₁-map, then, for all (x, b), (x′, b′) ∈ X ^h1,

$$\begin{array}{c}h\left(b\odot b^{\prime }\right)=h\left(b\right)\circ h\left(b^{\prime }\right)=f_{\mathrm{1}}\left(x\right)\circ f_{\mathrm{1}}\left(x^{\prime }\right)=f_{\mathrm{1}}\left(x\ast x^{\prime }\right).\end{array}$$ (3)

This implies

$$\begin{array}{c}(x,b)(\ast \times \odot )(x^{\prime },b^{\prime })=(x\ast x^{\prime },b\odot b^{\prime })\in X^{h\mathrm{1}}\end{array}$$ (4)

for all (x, b), (x′, b′) ∈ X ^h1. That is, (X ^h1|_τ12×υ, ∗ × ⊙) is an b-subspace of the bitopological semigroup ((X × O)_τ12×υ, ∗ × ⊙). Similarly, (X ^h2|_τ12×υ, ∗ × ⊙) is an b-subspace of the bitopological semigroup ((X × O)_τ12×υ, ∗ × ⊙).

Henceforth, in this paper, by J ₁ and J ₂, we mean the usual first and the second projection S-maps (or maps), respectively.

Theorem 6 . —

Let f ₁₂ : (X _τ12, ∗, X)→(O _ρ, ∘) be an C _ℵ-fibration and let h : (Q _υ, ⊙)→(O _ρ, ∘) be an S-map. Then the S-map f ^h1 : (X ^h1|_τ1×υ, ∗ × ⊙)→(Q _υ, ⊙) has the C _ℵ-fibration property by an S-map X ^h = X × id|_X^h2 such that f ^h1(x, b) = b for all (x, b) ∈ X ^h1.

Proof —

Since f ₁₂ is an c-map then, for all (x, b) ∈ X ^h2,

$$\begin{array}{c}{f}_{\mathrm{1}}\left(\mathcal{X}\left(x\right)\right)=f_{\mathrm{2}}\left(x\right)=h\left(b\right).\end{array}$$ (5)

That is, (X(x), b) ∈ X ^h1 for all (x, b) ∈ X ^h2. Hence, by the last lemma, X ^h is a well-defined S-map taking (X ^h2|_τ2×υ, ∗ × ⊙) into (X ^h1|_τ1×υ, ∗ × ⊙).

Now let (Y _ω, ⋆) ∈ ℵ and let g : (Y _ω, ⋆)→(X ^h2|_τ2×υ, ∗ × ⊙) and G : (Y _ω, ⋆) → P(Q _υ, ⊙) be two S-maps with G ₀ = f ^h1∘(X ^h∘g).

Take an S-map g′ = J ₁∘g : (Y _ω, ⋆)→(X _τ2, ∗) and an S-homotopy

$$\begin{array}{c}{G}^{\prime }=h\circ G:(Y_{\omega },\star )⟶P(O_{\rho },\circ ).\end{array}$$ (6)

We observe that

$$\begin{array}{c}{G}^{\prime }\left(y\right)\left(\mathrm{0}\right)=h\left[G\left(y\right)\left(\mathrm{0}\right)\right]=h\left[(f^{h\mathrm{1}}\circ {\mathcal{X}}^h)\left(g\left(y\right)\right)\right]\\ =h\left\{f^{h\mathrm{1}}[\mathcal{X}\left({\mathcal{J}}_{\mathrm{1}}\left(g\left(y\right)\right)\right),{\mathcal{J}}_{\mathrm{2}}\left(g\left(y\right)\right)]\right\}\\ =h\left[{\mathcal{J}}_{\mathrm{2}}\left(g\left(y\right)\right)\right]=f_{\mathrm{2}}\left[{\mathcal{J}}_{\mathrm{1}}\left(g\left(y\right)\right)\right]=f_{\mathrm{2}}\left(g^{\prime }\left(y\right)\right)\end{array}$$ (7)

for all y ∈ Y. That is, G ₀′ = f ₂∘g′. Since f ₁₂ is an C _ℵ-fibration, then there is an S-homotopy H′ : (Y _ω, ⋆) → P(X _τ1, ∗) such that H ₀′ = X∘g′ and f ₁∘H _t′ = G _t′ for all t ∈ I.

Define an S-homotopy H : (Y _ω, ⋆) → P(X ^h1|_τ1×υ, ∗ × ⊙) by

$$\begin{array}{c}H\left(y\right)\left(t\right)=[H^{\prime }\left(y\right)\left(t\right),G\left(y\right)\left(t\right)]\end{array}$$ (8)

for all y ∈ Y, t ∈ I. We observe that f ^h1∘H _t = G _t for all t ∈ I and

$$\begin{array}{c}H\left(y\right)\left(\mathrm{0}\right)=\left[H^{\prime }\left(y\right)\left(\mathrm{0}\right),G\left(y\right)\left(\mathrm{0}\right)\right]\\ =[\mathcal{X}\left(g^{\prime }\left(y\right)\right),(f^{h\mathrm{1}}\circ {\mathcal{X}}^h)\left(g\left(y\right)\right)]\\ =\left[\mathcal{X}\left({\mathcal{J}}_{\mathrm{1}}\left(g\left(y\right)\right)\right),{\mathcal{J}}_{\mathrm{2}}\left(g\left(y\right)\right)\right]\\ ={\mathcal{X}}^h[{\mathcal{J}}_{\mathrm{1}}\left(g\left(y\right)\right),{\mathcal{J}}_{\mathrm{2}}\left(g\left(y\right)\right)]\\ ={\mathcal{X}}^h\left(g\left(y\right)\right)=({\mathcal{X}}^h\circ g)\left(y\right)\end{array}$$ (9)

for all y ∈ Y. That is, H ₀ = X ^h∘g. Hence f ^h1 has the C _ℵ-fibration property by an S-map X ^h.

In the last theorem, if f ₁ = f ₂ (i.e., f ₁₂ is an Sp-map), let f = f ₁ = f ₂; then

$$\begin{array}{c}{\mathcal{X}}^h=\mathcal{X}\times id{|}_{X^h}:(X^h{|}_{{\tau }_{\mathrm{2}}\times \upsilon },\ast \times \odot )⟶(X^h{|}_{{\tau }_{\mathrm{1}}\times \upsilon },\ast \times \odot )\end{array}$$ (10)

is a well-defined S-map taking (X ^h|_τ2×υ, ∗ × ⊙) into (X ^h|_τ1×υ, ∗ × ⊙), where X ^h = X ^h1 = X ^h2. That is, the triple (X ^h|_τ12×υ, ∗ × ⊙, X ^h) is an c-bitopological semigroup, called a pullback c-bitopological semigroup of (X _τ12, ∗, X) induced from f ₁₂ by h. The pair

$$\begin{array}{c}{f}_{\mathrm{12}}^h=(f^h,f^h):{(X^h,{\tau }_{\mathrm{1}}\times \upsilon {|}_{X^h},{\tau }_{\mathrm{2}}\times \upsilon {|}_{X^h})}_{{\mathcal{X}}^h}⟶(Q_{\upsilon },\odot )\end{array}$$ (11)

which is given by f ^h(x, b) = b for all (x, b) ∈ X ^h is an c-map, called a pullback c-map of f ₁₂ induced by h. We observe that

$$\begin{array}{c}(f^h\circ {\mathcal{X}}^h)(x,b)=f^h(\mathcal{X}\left(x\right),b)=b=f^h(x,b)\end{array}$$ (12)

for all (x, b) ∈ X ^h.

Theorem 7 . —

Let f ₁₂ = (f, f):(X _τ12, ∗, X)→(O _ρ, ∘) be an C _ℵ-fibration and let h : (Q _υ, ⊙)→(O _ρ, ∘) be an S-map such that X ^h1∩X ^h2 ≠ ϕ. Then the pullback c-map f ₁₂ ^h of f ₁₂ induced by h is an C _ℵ-fibration.

Proof —

It is obvious by the last theorem and the second part in Definition 2.

4. The c-Lifting Functions

In this section, we define the path lifting property for c-maps by giving the concept of an c-lifting property and we show its role in satisfying the C _ℵ-fibration property.

Recall [5] that for an S-map f : (X _τ, ∗)→(O _ρ, ∘), the map: α → f∘α for all α ∈ P(X _τ) is an S-map from P(X _τ, ∗) into P(O _ρ, ∘), denoted by $\widehat{f}$. Then for every c-bitopological semigroup (X _τ12, ∗, X), $\widehat{\mathcal{X}}$ is an S-map from P(X _τ2, ∗) into P(X _τ1, ∗). That is, the triple $(P{(X)}_{{\tau }_{\mathrm{12}}^c},p(\ast ),\widehat{\mathcal{X}})$ is an c-bitopological semigroup where τ ₁ ^c and τ ₂ ^c are compact-open topologies on P(X) which are induced by τ ₁ and τ ₂, respectively. The shorter notion for this c-bitopological semigroup will be P(X _τ12, ∗, X).

For a map f : X _τ → O _ρ, by Δ(f), we mean the set

$$\begin{array}{c}\mathrm{\Delta }\left(f\right)=\{\left(x,\alpha \right)\in X_{\tau }\times P\left(O_{\rho }\right)\hspace{0.17em}\mid \hspace{0.17em}\alpha \left(\mathrm{0}\right)=f\left(x\right)\}.\end{array}$$ (13)

Proposition 8 . —

Let f : (X _τ, ∗)→(O _ρ, ∘) be an S-map. Then (Δ(f)|_τ×ρ^c, ∗ × p(∘)) is an S-subspace of an S-space ((X × P(O))_τ×ρ^c, ∗ × p(∘)), where ρ ^c is a compact-open topology on P(O) which is induced by ρ.

Proof —

It is clear that Δ(f)|_τ×ρ^c is a subspace of a space (X × P(O))_τ×ρ^c. We observe that, for all (x, α), (x′, α′) ∈ Δ(f),

$$\begin{array}{c}\left(\alpha p\left(\circ \right){\alpha }^{\prime }\right)\left(\mathrm{0}\right)=\alpha \left(\mathrm{0}\right)\circ {\alpha }^{\prime }\left(\mathrm{0}\right)=f\left(x\right)\circ f\left(x^{\prime }\right)=f\left(x\ast x^{\prime }\right).\end{array}$$ (14)

That is,

$$\begin{array}{c}(x,\alpha )\ast \times p(\circ )(x^{\prime },{\alpha }^{\prime })=(x\ast x^{\prime },\alpha p\left(\circ \right){\alpha }^{\prime })\in \mathrm{\Delta }\left(f\right).\end{array}$$ (15)

Hence (Δ(f)|_τ×ρ^c, ∗ × p(∘)) is an S-subspace of an S-space ((X × P(O))_τ×ρ^c, ∗ × p(∘)).

In the last theorem, the shorter notion for the S-space (Δ(f)|_τ×ρ^c, ∗ × p(∘)) will be Δ(f)|_τ×ρ^c.

Definition 9 . —

Let f ₁₂ : (X _τ12, ∗, X)→(O _ρ, ∘) be an c-map. An S-map

$$\begin{array}{c}L:\mathrm{\Delta }\left(f_{\mathrm{2}}\right){|}_{{\tau }_{\mathrm{2}}\times {\rho }^c}⟶P(X_{{\tau }_{\mathrm{1}}},\ast )\end{array}$$ (16)

from an S-space Δ(f ₂)|_τ2×ρ^c into P(X _τ1, ∗) is called an c-lifting function for an c-map f ₁₂ provided L satisfies the following:

1. L(x, α)(0) = X(x) for all (x, α) ∈ Δ(f ₂);

2. f ₁∘L(x, α) = α for all (x, α) ∈ Δ(f ₂).

And λ _f will be denoted to c-lifting function for an c-map f ₁₂, if it exists.

Example 10 . —

Let (X _τ12, ∗, X) be an c-bitopological semigroup. For every S-space (O _ρ, ∘), the Sp-map

$$\begin{array}{c}{f}_{\mathrm{12}}=(f,f):\left({\left(X\times O\right)}_{{\tau }_{\mathrm{12}}\times \rho },\ast \times \circ ,\mathcal{X}\times id\right)⟶(O_{\rho },\circ )\end{array}$$ (17)

is an c-map, where f(x, y) = y for all x ∈ X, y ∈ O. Note that

$$\begin{array}{c}[f\circ (\mathcal{X}\times \text{id})](x,y)=f(\mathcal{X}\left(x\right),y)=y=f(x,y)\end{array}$$ (18)

for all x ∈ X, y ∈ O. This c-map has an c-lifting function

$$\begin{array}{c}{\lambda }_f:\mathrm{\Delta }\left(f\right){|}_{({\tau }_{\mathrm{2}}\times \rho )\times {\rho }^c}⟶P({(X\times O)}_{{\tau }_{\mathrm{1}}\times \rho },{\tau }_{\mathrm{1}}\times \circ )\end{array}$$ (19)

which is given by

$$\begin{array}{c}{\lambda }_f\left(\left(x,b\right),\alpha \right)\left(t\right)=\left(\mathcal{X}\left(x\right),\alpha \left(t\right)\right)\hspace{1em}\forall \left(\left(x,b\right),\alpha \right)\in \mathrm{\Delta }\left(f\right),\hspace{0.17em}\hspace{0.17em}t\in I.\end{array}$$ (20)

Note that

$$\begin{array}{c}{\lambda }_f\left(\left(x,b\right),\alpha \right)\left(\mathrm{0}\right)=\left(\mathcal{X}\left(x\right),\alpha \left(\mathrm{0}\right)\right)\\ =\left(\mathcal{X}\left(x\right),f\left(x,b\right)\right)\\ =\left(\mathcal{X}\left(x\right),b\right)\\ =\left(\mathcal{X}\times \text{id}\right)\left(x,b\right),\\ [f\circ {\lambda }_f((x,b),\alpha )]\left(t\right)=f(\mathcal{X}\left(x\right),\alpha \left(t\right))=\alpha \left(t\right)\end{array}$$ (21)

for all ((x, b), α) ∈ Δ(f), t ∈ I.

The following theorem clarifies the existence property for c-lifting function in C _ℵ-fibration theory. That is, it clarifies that the existence of c-lifting function for any C _ℵ-fibration is necessary and sufficient condition.

Theorem 11 . —

An c-map f ₁₂ : (X _τ12, ∗, X)→(O _ρ, ∘) is an C _ℵ-fibration if and only if there exists an c-lifting function for f ₁₂.

Proof —

Suppose that f ₁₂ is an C _ℵ-fibration. Take Δ(f ₂)|_τ2×ρ^c ∈ ℵ. Define two S-maps

$$\begin{array}{c}{g:\mathrm{\Delta }\left(f_{\mathrm{2}}\right)|}_{{\tau }_{\mathrm{2}}\times {\rho }^c}⟶\left(X_{{\tau }_{\mathrm{2}}},\ast \right),\\ {G:\mathrm{\Delta }\left(f_{\mathrm{2}}\right)|}_{{\tau }_{\mathrm{2}}\times {\rho }^c}⟶P(O_{\rho },\circ )\end{array}$$ (22)

by g(x, α) = x and G(x, α) = α for all (x, α) ∈ Δ(f ₂), respectively. We observe that

$$\begin{array}{c}G(x,\alpha )\left(\mathrm{0}\right)=\alpha \left(\mathrm{0}\right)=f_{\mathrm{2}}\left(x\right)=(f_{\mathrm{2}}\circ g)(x,\alpha ).\end{array}$$ (23)

Since f ₁₂ is an C _ℵ-fibration, then there exists an S-homotopy H : Δ(f ₂)|_τ2×ρ^c → P(X _τ1, ∗) such that H ₀ = X∘g and f ₁∘H _t = G _t for all t ∈ I. Define an S-map

$$\begin{array}{c}{{\lambda }_f:\mathrm{\Delta }\left(f_{\mathrm{2}}\right)|}_{{\tau }_{\mathrm{2}}\times {\rho }^c}⟶P(X_{{\tau }_{\mathrm{1}}},\ast )\end{array}$$ (24)

by

$$\begin{array}{c}{\lambda }_f(x,\alpha )\left(t\right)=H(x,\alpha )\left(t\right)\hspace{1em}\forall (x,\alpha )\in \mathrm{\Delta }\left(f_{\mathrm{2}}\right).\end{array}$$ (25)

We observe that, for all (x, α) ∈ Δ(f ₂),

$$\begin{array}{c}{\lambda }_f(x,\alpha )\left(\mathrm{0}\right)=H(x,\alpha )\left(\mathrm{0}\right)=(\mathcal{X}\circ g)(x,\alpha )=\mathcal{X}\left(x\right)\\ f_{\mathrm{1}}\circ {\lambda }_f(x,\alpha )=f_{\mathrm{1}}\circ H(x,\alpha )=G(x,\alpha )=\alpha .\end{array}$$ (26)

That is, λ _f is an c-lifting function for f ₁₂.

Conversely, suppose that there exists an c-lifting function λ _f for f ₁₂. Let (Y _ω, ⋆) ∈ ℵ and let g : (Y _ω, ⋆)→(X _τ2, ∗) and G : (Y _ω, ⋆) → P(O _ρ, ∘) be two given S-maps with G ₀ = f ₂∘g. Define an S-homotopy H : (Y _ω, ⋆) → P(X _τ1, ∗) by

$$\begin{array}{c}H\left(y\right)\left(t\right)={\lambda }_f[g\left(y\right),G\left(y\right)]\left(t\right)\hspace{1em}\forall y\in Y,\hspace{0.17em}\hspace{0.17em}t\in I.\end{array}$$ (27)

We observe that

$$\begin{array}{c}H\left(y\right)\left(\mathrm{0}\right)={\lambda }_f\left[g\left(y\right),G\left(y\right)\right]\left(\mathrm{0}\right)=\left(\mathcal{X}\circ g\right)\left(y\right),\\ f_{\mathrm{1}}\left[H\left(y\right)\left(t\right)\right]=f_{\mathrm{1}}\left[{\lambda }_f[g\left(y\right),G\left(y\right)]\left(t\right)\right]=G\left(y\right)\left(t\right)\end{array}$$ (28)

for all y ∈ Y, t ∈ I. That is, H ₀ = X∘g and f ₁∘H _t = G _t for all t ∈ I. Hence f ₁₂ is an C _ℵ-fibration.

Theorem 12 . —

Let f ₁₂ : (X _τ12, ∗, X)→(O _ρ, ∘) be an C _ℵ-fibration. Then the c-map

$$\begin{array}{c}{\widehat{f}}_{\mathrm{12}}=({\widehat{f}}_{\mathrm{1}},{\widehat{f}}_{\mathrm{2}}):P(X_{{\tau }_{\mathrm{12}}},\ast ,\mathcal{X})⟶P(O_{\rho },\circ )\end{array}$$ (29)

is an C _ℵ-fibration.

Proof —

Since f ₁₂ is an C _ℵ-fibration, then there exists c-lifting function

$$\begin{array}{c}{{\lambda }_f:\mathrm{\Delta }\left(f_{\mathrm{2}}\right)|}_{{\tau }_{\mathrm{2}}\times {\rho }^c}⟶P(X_{{\tau }_{\mathrm{1}}},\ast )\end{array}$$ (30)

for f ₁₂ such that

$$\begin{array}{c}{\lambda }_f\left(x,\alpha \right)\left(\mathrm{0}\right)=\mathcal{X}\left(x\right),\\ f_{\mathrm{1}}\circ {\lambda }_f(x,\alpha )=\alpha \end{array}$$ (31)

for all (x, α) ∈ Δ(f ₂). Let (Y _ω, ⋆) ∈ ℵ and let g : (Y _ω, ⋆) → P(X _τ2, ∗) and G : (Y _ω, ⋆) → P[P(O)_ρ^c, p(∘)] be two given S-maps with

$$\begin{array}{c}\left[G\left(y\right)\left(s\right)\right]\left(\mathrm{0}\right)=({\widehat{f}}_{\mathrm{2}}\circ g)\left(y\right)\left(s\right)\end{array}$$ (32)

for all y ∈ Y, s ∈ I, where ρ ^c is a compact-open topology on P(O) which is induced by ρ. Define an S-homotopy H : (Y _ω, ⋆) → P[P(X)_τ1^c, p(∗)] by

$$\begin{array}{c}\left[H\left(y\right)\left(s\right)\right]\left(t\right)={\lambda }_f\left[g\left(y\right)\left(s\right),G\left(y\right)\left(s\right)\right]\left(t\right)\forall y\in Y,\hspace{0.17em}\hspace{0.17em}s,t\in I.\end{array}$$ (33)

We observe that

$$\begin{array}{c}\left[H\left(y\right)\left(s\right)\right]\left(\mathrm{0}\right)={\lambda }_f\left[g\left(y\right)\left(s\right),G\left(y\right)\left(s\right)\right]\left(\mathrm{0}\right)\\ =\mathcal{X}\left[g\left(y\right)\left(s\right)\right]=\left(\widehat{\mathcal{X}}\circ g\right)\left(y\right)\left(s\right),\\ \left({\widehat{f}}_{\mathrm{1}}\circ H_r\right)\left(y\right)\left(t\right)=\left(f_{\mathrm{1}}\circ {\lambda }_f\left[g\left(y\right)\left(s\right),G\left(y\right)\left(s\right)\right]\right)\left(t\right)\\ =\left[G\left(y\right)\left(s\right)\right]\left(t\right),\end{array}$$ (34)

for all y ∈ Y, s, t ∈ I. That is, $H_{\mathrm{0}}=\widehat{\mathcal{X}}\circ g$ and ${\widehat{f}}_{\mathrm{1}}\circ H_s=G_s$ for all s ∈ I. Hence ${\widehat{f}}_{\mathrm{12}}$ is an C _ℵ-fibration.

An c-lifting function λ _f is called regular if for every x ∈ X _τ2, ${\lambda }_f(x,f_{\mathrm{2}}\circ \tilde{x})=\widetilde{\mathcal{X}(x)}$, where $\tilde{x}$ is the constant path in X _τ2 (i.e., $\tilde{x}(t)=x$), similar for $\widetilde{\mathcal{X}(x)}$. An C _ℵ-fibration f ₁₂ is called regular if it has regular c-lifting function.

Example 13 . —

In Example 10, the c-lifting function λ _f which is given by

$$\begin{array}{c}{\lambda }_f\left(\left(x,b\right),\alpha \right)\left(t\right)=\left(\mathcal{X}\left(x\right),\alpha \left(t\right)\right)\hspace{1em}\forall \left(\left(x,b\right),\alpha \right)\in \mathrm{\Delta }\left(f\right),\hspace{0.17em}\hspace{0.17em}t\in I\end{array}$$ (35)

is regular. Note that, for every (x, b)∈(X × O)_τ2,

$$\begin{array}{c}{\lambda }_f\left(\left(x,b\right),f_{\mathrm{2}}\circ \widetilde{\left(x,b\right)}\right)\left(t\right)=\left(\mathcal{X}\left(x\right),\left(f_{\mathrm{2}}\circ \widetilde{\left(x,b\right)}\right)\left(t\right)\right)\\ =(\mathcal{X}\left(x\right),f_{\mathrm{2}}(x,b))\\ =\left(\mathcal{X}\left(x\right),b\right)=\left(\mathcal{X}\times \text{id}\right)\left(x,b\right)\\ \hspace{0.17em}=\widetilde{(\mathcal{X}\times \text{id})(x,b)}\left(t\right)\end{array}$$ (36)

for all t ∈ I.

The following theorem is an analogue of results of Fadell in Hurewicz fibration theory [7].

Theorem 14 . —

Let f ₁₂ : (X _τ12, ∗, X)→(O _ρ, ∘) be a regular C _ℵ-fibration and let

$$\begin{array}{c}{M}_i:P(X_{{\tau }_i},\ast )⟶{\mathrm{\Delta }\left(f_i\right)|}_{{\tau }_i\times {\rho }^c}\end{array}$$ (37)

be an S-map defined by M _i(α) = (α(0), f _i∘α) for all α ∈ P(X _τi) where i = 1,2. Then

• (1) M ₁∘λ _f = X × id|_Δ(f2);
• (2)
  ${\lambda }_f\circ M_{\mathrm{2}}\hspace{0.17em}{\simeq }_s\hspace{0.17em}\widehat{\mathcal{X}}$ preserving projection. That is, there is an S-homotopy

  $$\begin{array}{c}H:P(X_{{\tau }_{\mathrm{2}}},\ast )⟶P[P{\left(X\right)}_{{\tau }_{\mathrm{1}}^c},p\left(\ast \right)]\end{array}$$ (38)

  between two S-maps λ _f∘M ₂ and $\widehat{\mathcal{X}}$ such that f ₁[(H(α)(s))(t)] = f ₂(α(t)) for all t, s ∈ I, α ∈ P(X _τ2).

Proof —

For the first part, we observe that, for every (x, α) ∈ Δ(f ₂),

$$\begin{array}{c}\left(M_{\mathrm{1}}\circ {\lambda }_f\right)\left(x,\alpha \right)=M_{\mathrm{1}}\left[{\lambda }_f\left(x,\alpha \right)\right]\\ =({\lambda }_f(x,\alpha )\left(\mathrm{0}\right),f_{\mathrm{1}}\circ {\lambda }_f(x,\alpha ))\\ =(\mathcal{X}\left(x\right),\alpha )=(\mathcal{X}\times \text{id})(x,\alpha ).\end{array}$$ (39)

That is, M ₁∘λ _f = X × id|_Δ(f2).

For the second part, for α ∈ P(X _τ2) and s ∈ I, define a path β ^1−s ∈ P(O _ρ) by

$$\begin{array}{c}{\beta }^{\mathrm{1}-s}\left(t\right)=\{\begin{array}{cc}{f}_{\mathrm{2}}\left(\alpha (s+t)\right),\hfill & \text{for}\hspace{0.17em}\hspace{0.17em}\mathrm{0}\le t\le \mathrm{1}-s,\hfill \\ f_{\mathrm{2}}\left(\alpha \left(\mathrm{1}\right)\right),\hfill & \text{for}\hspace{0.17em}\hspace{0.17em}\mathrm{1}-s\le t\le \mathrm{1}.\hfill \end{array}\end{array}$$ (40)

By the regularity of λ _f, we can define an S-homotopy H : P(X _τ2, ∗) → P[P(X)_τ1^c, p(∗)] by

$$\begin{array}{c}\left[H\left(\alpha \right)\left(s\right)\right]\left(t\right)=\{\begin{array}{cc}\mathcal{X}\left[\alpha \left(t\right)\right],\hfill & \text{for}\hspace{0.17em}\hspace{0.17em}\mathrm{0}\le t\le s,\hfill \\ {\lambda }_f(\alpha \left(s\right),{\beta }^{\mathrm{1}-s})(t-s),\hfill & \text{for}\hspace{0.17em}\hspace{0.17em}s\le t\le \mathrm{1},\hfill \end{array}\end{array}$$ (41)

for all s ∈ I, α ∈ P(X _τ2). Then

$$\begin{array}{c}\left[H\left(\alpha \right)\left(\mathrm{0}\right)\right]\left(t\right)={\lambda }_f\left(\alpha \left(\mathrm{0}\right),{\beta }^{\mathrm{1}}\right)\left(t\right)\\ ={\lambda }_f(\alpha \left(\mathrm{0}\right),\beta )\left(t\right)\\ ={\lambda }_f(\alpha \left(\mathrm{0}\right),f_{\mathrm{2}}\circ \alpha )\left(t\right)\\ =\left({\lambda }_f\circ M_{\mathrm{2}}\right)\left(\alpha \right)\left(t\right),\\ \left[H\left(\alpha \right)\left(\mathrm{1}\right)\right]\left(t\right)=\mathcal{X}\left[\alpha \left(t\right)\right]=\widehat{\mathcal{X}}\left(\alpha \right)\left(t\right)\end{array}$$ (42)

for all α ∈ P(X _τ2), t ∈ I. That is, ${\lambda }_f\circ M_{\mathrm{2}}\hspace{0.17em}{\simeq }_s\hspace{0.17em}\widehat{\mathcal{X}}$. Also we get that

$$\begin{array}{c}\left[f_{\mathrm{1}}\circ \left(H\left(\alpha \right)\left(s\right)\right)\right]\left(t\right)\\ \hspace{1em}\hspace{1em}=\{\begin{array}{cc}{f}_{\mathrm{1}}\left(\mathcal{X}\left[\alpha \left(t\right)\right]\right),\hfill & \text{for}\hspace{0.17em}\hspace{0.17em}\mathrm{0}\le t\le s,\hfill \\ f_{\mathrm{1}}\left[{\lambda }_f(\alpha \left(s\right),{\beta }^{\mathrm{1}-s})(t-s)\right],\hfill & \text{for}\hspace{0.17em}\hspace{0.17em}s\le t\le \mathrm{1};\hfill \end{array}\\ \hspace{1em}\hspace{1em}=\{\begin{array}{cc}{f}_{\mathrm{2}}\left(\alpha \left(t\right)\right),\hfill & \text{for}\hspace{0.17em}\hspace{0.17em}\mathrm{0}\le t\le s,\hfill \\ {\beta }^{\mathrm{1}-s}(t-s),\hfill & \text{for}\hspace{0.17em}\hspace{0.17em}s\le t\le \mathrm{1};\hfill \end{array}\\ \hspace{1em}\hspace{1em}=\{\begin{array}{cc}{f}_{\mathrm{2}}\left(\alpha \left(t\right)\right),\hfill & \text{for}\hspace{0.17em}\hspace{0.17em}\mathrm{0}\le t\le s,\hfill \\ f_2\left(\alpha (s+t-s)\right),\hfill & \text{for}\hspace{0.17em}\hspace{0.17em}s\le t\le \mathrm{1};\hfill \end{array}\\ \hspace{1em}\hspace{1em}=\{\begin{array}{cc}{f}_{\mathrm{2}}\left(\alpha \left(t\right)\right),\hfill & \text{for}\hspace{0.17em}\hspace{0.17em}\mathrm{0}\le t\le s,\hfill \\ f_{\mathrm{2}}\left(\alpha \left(t\right)\right),\hfill & \text{for}\hspace{0.17em}\hspace{0.17em}s\le t\le \mathrm{1};\hfill \end{array}\\ \hspace{1em}\hspace{1em}=f_{\mathrm{2}}\left(\alpha \left(t\right)\right),\end{array}$$ (43)

for all s, t ∈ I, α ∈ P(X _τ2). Hence ${\lambda }_f\circ M_{\mathrm{2}}\hspace{0.17em}{\simeq }_s\hspace{0.17em}\widehat{\mathcal{X}}$ preserving projection.

5. Approximate Fibrations

Coram and Duvall [2] introduced approximate fibrations as a generalization of cell-like maps [3] and showed that the uniform limit of a sequence of Hurewicz fibrations is an approximate fibration. A map f : X _τ → O _ρ of compact metrizable spaces X _τ and O _ρ is called an approximate fibration if, for every space Y _ω and for given ϵ > 0, there exists δ > 0 such that whenever g : Y _ω → X _τ and H : Y _ω × I → O _ρ are maps with d[H(y, 0), (f∘g)(y)] < δ, then there is homotopy G : Y _ω × I → X _τ such that G ₀ = g and

$$\begin{array}{c}d[H(y,t),(f\circ G)(y,t)]<ϵ\hspace{1em}\forall y\in Y,\hspace{0.17em}t\in I.\end{array}$$ (44)

One notable exception is that the pullback of approximate fibration need not be an approximate fibration.

The following theorem shows the role of the C _Nπ-fibration property in inducing an approximate fibration property.

For an S-map f : (X _τ, π)→(O _ρ, π) with metrizable spaces X _τ and O _ρ, by d _τ and d _ρ, we mean the metric functions on X and O, respectively; by X × O we mean the product metrizable space of X _τ and O _ρ with a metric function

$$\begin{array}{c}d((x,b),(x^{\prime },b^{\prime }))=\max \{d_{\tau }(x,x^{\prime }),d_{\rho }(b,b^{\prime })\};\end{array}$$ (45)

by G ^f we mean the graph of f (i.e., G ^f = {(x, f(x)) : x ∈ X}) which is an S-subspace of ((X × O)_τ×ρ, π); for a positive integer n > 0, by G ⁿ(f), we mean the (1/n)-neighborhood of G ^f in a metrizable space X × O which is also S-subspace of ((X × O)_τ×ρ, π).

Theorem 15 . —

Let f : X _τ → O _ρ be a map with compact metrizable spaces X _τ and O _ρ. Then f is an approximate fibration if and only if, for every positive integer n > 0, there exists a positive integer m ≥ n such that the S-map f _n : (G ⁿ(f), π)→(O _ρ, π) has the C _Nπ-fibration property by the inclusion S-map I _m ⁿ : (G ^m(f), π)→(G ⁿ(f), π), where f _n(x, b) = b for all (x, b) ∈ G ⁿ(f).

Proof —

Let n be any positive integer. For ϵ = 1/n > 0, let δ be given in the definition of approximate fibration. Since δ/2 > 0 and f is a continuous function, then let δ′ be chosen such that if x, x′ ∈ X and d _τ(x, x′) < δ′, then d _ρ(f(x), f(x′)) < ϵ/2. Choose a positive integer m⩾n, such that 1/m ≤ δ′, δ/2.

Now let (Y _ω, π) ∈ N _π and let g : (Y _ω, π)→(G ^m(f), π) and G : (Y _ω, π) → P(O _ρ, π) be two given S-maps with G ₀ = f _n∘(I _m ⁿ∘g). Define a map g′ : Y _ω → X _τ by g′(y) = J ₁[g(y)] and a homotopy G′ : Y _ω × I → O _ρ by G′(y, t) = G(y)(t) for all y ∈ Y and t ∈ I. We get that g(y) = (g′(y), G′(y, 0)) for all y ∈ Y. Since g(y) ∈ G ^m(f), then there exists x ∈ X such that

$$\begin{array}{c}d[(x,f\left(x\right)),g\left(y\right)]<\frac{\mathrm{1}}{m}.\end{array}$$ (46)

Then

$$\begin{array}{c}{d}_{\tau }\left(x,g^{\prime }\left(y\right)\right)<\frac{\mathrm{1}}{m}⩽{\delta }^{\prime },\\ d_{\rho }(f\left(x\right),G^{\prime }(y,\mathrm{0}))<\frac{\mathrm{1}}{m}⩽\frac{\delta }{\mathrm{2}},\\ d_{\rho }(f\left(x\right),f\left(g^{\prime }\left(y\right)\right))<\frac{\mathrm{1}}{m}⩽\frac{\delta }{\mathrm{2}}\end{array}$$ (47)

for all y ∈ Y. This implies

$$\begin{array}{c}{d}_{\rho }\left(f\left(g^{\prime }\left(y\right)\right),G^{\prime }\left(y,\mathrm{0}\right)\right)\\ \hspace{1em}\hspace{1em}⩽d_{\rho }\left(f\left(g^{\prime }\left(y\right)\right),f\left(x\right)\right)+d_{\rho }\left(f\left(x\right),G^{\prime }\left(y,\mathrm{0}\right)\right)\\ \hspace{1em}\hspace{1em}<\delta \end{array}$$ (48)

for all y ∈ Y. Hence, since f is an approximate fibration, there exists a homotopy H′ : Y _ω × I → X _τ such that H ₀′ = g′ and

$$\begin{array}{c}{d}_{\tau }(G^{\prime }(y,t),(f\circ H_t^{\prime })\left(y\right))<ϵ\end{array}$$ (49)

for all y ∈ Y, t ∈ I. Define an S-homotopy H : (Y _ω, π) → P(G ⁿ(f), π) by

$$\begin{array}{c}H\left(y\right)\left(t\right)=(H^{\prime }(y,t),G\left(y\right)\left(t\right))\hspace{1em}\forall y\in Y,\hspace{0.17em}\hspace{0.17em}t\in I.\end{array}$$ (50)

Then we get that

$$\begin{array}{c}H\left(y\right)\left(\mathrm{0}\right)=\left(H^{\prime }\left(y,\mathrm{0}\right),G\left(y\right)\left(\mathrm{0}\right)\right)=\left(g^{\prime }\left(y\right),G\left(y\right)\left(\mathrm{0}\right)\right)\\ =g\left(y\right)=({\mathcal{I}}_m^n\circ g)\left(y\right)\end{array}$$ (51)

for all y ∈ Y and f _n∘H _t = G _t for all t ∈ I. Hence f _n has the C _Nπ-fibration property by I _m ⁿ.

Conversely, let ϵ > 0 be given. Since f is a continuous function, then let δ′ be chosen such that if x, x′ ∈ X and d _τ(x, x′) < δ′, then d _ρ(f(x), f(x′)) < ϵ/2. Choose a positive integer n > 0 such that 1/n ≤ δ′, ϵ/2. By hypothesis, there exists a positive integer m ≥ n such that f _n has the C _Nπ-fibration property by I _m ⁿ.

Take δ = 1/m. Let Y _ω be any space and let g : Y _ω → X _τ and G : Y _ω × I → O _ρ be two given maps with

$$\begin{array}{c}{d}_{\rho }[G(y,\mathrm{0}),(f\circ g)\left(y\right)]<\delta \end{array}$$ (52)

for all y ∈ Y. Define an S-map g′ : (Y _ω, π)→(G ^m(f), π) by g′(y) = (g(y), G(y, 0)) and an S-homotopy G′ : (Y _ω, π) → P(O _ρ, ı) by G′(y)(t) = G(y, t) for all y ∈ Y and t ∈ I. Since G ₀′ = f _n∘(I _m ⁿ∘g′), then there exists an S-homotopy F : (Y _ω, π) → P(G ⁿ(f), π) such that F ₀ = I _m ⁿ∘g′ and f _n∘F _t = G _t′ for all t ∈ I. By the last part, we can define a homotopy H : Y _ω × I → X _τ by

$$\begin{array}{c}H(y,t)={\mathcal{J}}_{\mathrm{1}}\left[F\left(y\right)\left(t\right)\right]\hspace{1em}\forall y\in Y,\hspace{0.17em}\hspace{0.17em}t\in I.\end{array}$$ (53)

We get that F(y)(t) = (H(y, t), G(y, t)). Since F(y)(t) ∈ G ⁿ(f), then there exists x ∈ X such that

$$\begin{array}{c}d[(x,f\left(x\right)),F\left(y\right)\left(t\right)]<\frac{\mathrm{1}}{n}.\end{array}$$ (54)

Then

$$\begin{array}{c}{d}_{\tau }\left(x,H\left(y,t\right)\right)<\frac{\mathrm{1}}{n}⩽{\delta }^{\prime },\hspace{1em}{d}_{\rho }(f\left(x\right),G(y,t))<\frac{\mathrm{1}}{n}⩽\frac{ϵ}{\mathrm{2}},\\ d_{\rho }\left[f\left(x\right),f\left(H\left(y,t\right)\right)\right]<\frac{\mathrm{1}}{n}⩽\frac{ϵ}{\mathrm{2}}.\end{array}$$ (55)

This implies

$$\begin{array}{c}{d}_{\rho }\left[G\left(y,t\right),f\left(H\left(y,t\right)\right)\right]\\ \hspace{1em}\hspace{1em}⩽d_{\rho }[f\left(H(y,t)\right),f\left(x\right)]+d_{\rho }(f\left(x\right),G(y,t))<ϵ\end{array}$$ (56)

for all y ∈ Y, t ∈ I. Hence f is an approximate fibration.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.