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

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.