Homotopy, homology, and persistent homology using closure spaces

TL;DR

This paper extends persistent homology to filtrations of closure spaces, including metric spaces and graphs, introducing new homotopy and homology theories and establishing stability of persistence modules.

Contribution

It develops six homotopy theories and nine homology theories for closure spaces, generalizing persistent homology beyond traditional topological spaces.

Findings

Introduces new homotopy and homology theories for closure spaces.

Extends Gromov-Hausdorff distance to closure space filtrations.

Proves stability of persistence modules derived from homotopy-invariant functors.

Abstract

We develop persistent homology in the setting of filtrations of (Cech) closure spaces. Examples of filtrations of closure spaces include metric spaces, weighted graphs, weighted directed graphs, and filtrations of topological spaces. We use various products and intervals for closure spaces to obtain six homotopy theories, six cubical singular homology theories, and three simplicial singular homology theories. Applied to filtrations of closure spaces, these homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance from metric spaces to filtrations of closure spaces and use it to prove that any persistence module obtained from a homotopy-invariant functor on closure spaces is stable.