Homogeneous spaces not separated by arcs

TL;DR

This paper improves previous results by showing that in certain homogeneous metric spaces, regions are not separated by arcs, even under weaker conditions and with only one endpoint of the arc in the interior.

Contribution

The authors extend prior work by weakening the homogeneity condition and considering arcs with only one endpoint in the interior of the region.

Findings

Regions in homogeneous locally compact metric spaces of dimension ≥ 2 are not separated by arcs.

The result holds even when only one arc endpoint is in the interior.

The improvement replaces strong local homogeneity with homogeneity.

Abstract

It was shown by van Mill and Valov that regions in strongly locally homogeneous locally compact metric spaces of dimension $\ge 2$ are not separated by arcs. We improve this result by replacing strong local homogeneity with homogeneity. Moreover, we prove the result for the case when only one end point of an arc is in the interior of the region.