A metric space (X, d) is called an M-space if for every x and y in X and for every r 6 [0, A] we have B[x, r] Cl B[y, A — r] = {2} for some z € X, where A = d(x, y). It is the object of this paper to study M-spaces in terms of proximinality properties of certain sets. 0. Introduction. Let (X, d) be a metric space, and G be a closed subset of X. For x E X, let p(x,G) = inf{d(x, y) : y E G}. If t...