Boundedly connected sets and the distance to the intersection of two sets

We prove that if (X, d) is a metric space, C is a closed subset of X and x ∈ X, then the distance of x to R ∩ S agrees with the maximum of the distances of x to R and S, for every closed subsets R,S ⊂ C such that C = R ∪ S, if and only if C is x-boundedly connected. © 2006 Elsevier Inc. All rights reserved