Separation of homogeneous connected locally compact spaces

TL;DR

This paper proves a topological separation property in homogeneous locally compact spaces, extending known results by analyzing acyclic subsets and local bases, with implications for understanding space connectivity.

Contribution

It introduces a new separation theorem involving acyclic subsets and local bases in homogeneous locally compact spaces, generalizing previous results.

Findings

Any region in a homogeneous space cannot be separated by certain acyclic subsets.

The result applies to strongly locally homogeneous spaces with simpler conditions.

It unifies and extends existing theorems on space separation in topology.

Abstract

We prove that any region $\Gamma$ in a homogeneous $n$-dimensional and locally compact separable metric space $X$, where $n\geq 2$, cannot be irreducibly separated by a closed $(n-1)$-dimensional subset $C$ with the following property: $C$ is acyclic in dimension $n-1$ and there is a point $b\in C\cap\Gamma$ having a special local base $\mathcal B_C^b$ in $C$ such that the boundary of each $U\in\mathcal B_C^b$ is acyclic in dimension $n-2$. In case $X$ is strongly locally homogeneous, it suffices to have a point $b\in C\cap\Gamma$ with an ordinary base $\mathcal B_C^b$ satisfying the above condition. The acyclicity means triviality of the corresponding \v{C}ech cohomology groups. This implies all known results concerning the separation of regions in homogeneous connected locally compact spaces.