A homological nerve theorem for open covers

TL;DR

This paper extends a homological nerve theorem from finite simplicial complexes to arbitrary topological spaces with open covers, addressing a question about homology groups of Vietoris metric thickenings.

Contribution

It generalizes a known homological nerve theorem to open covers of any topological space, broadening its applicability.

Findings

The theorem now applies to open covers of arbitrary topological spaces.

It confirms the homology groups of Vietoris metric thickenings can be analyzed using this generalized theorem.

Abstract

In this note we show that a particular homological nerve theorem, which was originally proved for a finite cover of a simplicial complex by subcomplexes, also holds for an open cover of an arbitrary topological space. The motivation for this is to affirmatively answer a question about the homology groups of Vietoris metric thickenings.