Sufficent Conditions for the preservation of Polygonal-Connectedness in an arbitrary normed space

TL;DR

This paper establishes sufficient conditions under which polygonal-connectedness is preserved in normed spaces when removing certain discrete subsets, extending understanding of connectivity properties in geometric analysis.

Contribution

It provides new criteria ensuring polygonal-connectedness remains intact after removing specific discrete subsets from a polygonally-connected set in a normed space.

Findings

Polygonal-connectedness is preserved under certain conditions.

Conditions involve the nature of the removed discrete subsets.

Results apply to arbitrary normed spaces.

Abstract

In this article we prove that if $ X $ is a normed space and $ U $ is a polygonally-connected subset of $ X $ with $M:=\{S_i:\ i\in I\}\subset \mathcal{P}\left( U\right) $, a non-empty arbitrary family of discrete, non-empty subsets of $ U, $ then the property of polygonal-connectedness is also preserved in the resulting set $ U\setminus\left( \bigcup_{i\in I} S_i\right),$ under appropriate conditions.