Mazurkiewicz manifolds and homogeneity

TL;DR

This paper proves that in homogeneous, locally compact, locally connected metric spaces, no region can be separated by an $F_\sigma$-subset of lower dimension, with implications across various topological dimensions.

Contribution

It establishes a new topological property linking homogeneity, local compactness, and the inability of lower-dimensional sets to cut regions.

Findings

No region in the space can be separated by an $F_\sigma$-subset of smaller dimension.

The result applies to spaces with finite or infinite topological dimensions.

The theorem extends to various classes of metrizable spaces.

Abstract

It is proved that no region of a homogeneous locally compact, locally connected metric space can be cut by an $F_\sigma$-subset of a "smaller" dimension. The result applies to different finite or infinite topological dimensions of metrizable spaces.