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

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.