Alexander-Spanier cohomology and boundary of a domain

TL;DR

This paper explores the relationship between the boundary connectivity of a domain and its topological properties, extending previous results from compact manifolds to more general locally connected metric spaces.

Contribution

It generalizes the equivalence between boundary disconnectedness and trivial first cohomology from compact manifolds to locally connected metric spaces.

Findings

A domain cuts the ambient space if and only if its boundary is disconnected.

For compact manifolds, this property is equivalent to trivial 1-cohomology.

The results are extended to locally connected metric spaces.

Abstract

We investigate a property: a domain cuts the ambient space if and only if it's boundary is disconnected. Previously, we had shown that for compact manifolds this property is equivalent to the manifold having trivial 1-cohomology. We now generalize this discussion to locally connected metric spaces.