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:=ᵢ:\ i∈\1,2,,k\\⊂ int(U), then the property of path-connectedness is also preserved in the resulting set U 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.