Sufficient 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 := {xi : 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 provide path-connectedness for domains with holes.