Sufficent Conditions for the preservation of Path-Connectedness in an arbitrary metric space

TL;DR

This paper establishes conditions under which path-connectedness is preserved in metric spaces when removing certain subsets, including countably infinite sets, especially in domains with holes.

Contribution

It provides sufficient conditions for preserving path-connectedness in metric spaces after removing specific subsets, extending to countably infinite cases.

Findings

Path-connectedness is preserved when removing finite subsets with path-connected boundaries.

The results extend to countably infinite subsets under certain conditions.

Applications include maintaining connectivity in domains with holes.

Abstract

It is proven that if $ (X,d) $ is an arbitrary metric space and $ U $ is a path-connected subset of $ X $ with $M:=\{x_i:\ i\in\{1,2,\dots,k\}\}\subset int(U) $, then the property of path-connectedness is also preserved in the resulting set $ U\setminus M, $ provided that the boundary of each open ball of X is a non-empty and path-connected set. Moreover, under appropriate conditions we extend the above result in the case where the set $ M $ is countably infinite. As a consequence these results maintain path-connectedness for domains with holes.