Comparison of persistent singular and \v{C}ech homology for locally connected filtrations

TL;DR

This paper demonstrates that for certain well-behaved filtrations, persistent singular homology and persistent Čech homology are essentially equivalent, with zero interleaving distance, highlighting their close relationship in topological data analysis.

Contribution

It establishes the equality of persistent singular and Čech homology in locally connected filtrations, extending understanding of their relationship in topological data analysis.

Findings

Interleaving distance between the two homologies is zero.

Results apply to filtrations of paracompact Hausdorff spaces.

Enhances theoretical foundation of persistent homology methods.

Abstract

We show that the interleaving distance between the persistent singular homology and the persistent \v{C}ech homology of a homologically locally connected filtration consisting of paracompact Hausdorff spaces is 0.