Sufficient conditions for the preservation of polygonal-connectedness in an arbitrary normed space

In this article we prove that if \( (X,\left\Vert \cdot \right\Vert _X) \) $(X,{∥·∥}_X)$ is a normed space and U is a polygonally-connected subset of X with \(M:=\{S_i:\ i\in I\}\subset \mathscr {P}\left( U\right) , \) $M:=\{S_i:i\in I\}\subset \mathcal{P}\left(U\right),$ 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 \( U\setminus \left( \bigcup _{i\in I} S_i\right) ,\) $U\backslash \left({\bigcup }_{i\in I},S_i\right),$ under appropriate conditions.