Figure 1.

Example of persistent homology for a point cloud. (a) A finite set of points in ${\mathbb{R}}^2$ (for $ϵ=0$) and a nested sequence of spaces obtained from it (from $ϵ=0$ to $ϵ=2.1$). (b) Barcode for the nested sequence of spaces illustrated in (a). Solid lines represent the lifetime of components, and dashed lines represent the lifetime of holes.