The unique midpoint property of a subspace of the real line

A metric space X is said to have the unique midpoint property (UMP) if there is a metric d on X which induces the topology of X and such that for each pair of distinct points x,y∈X, there is one and only one point p∈X with d(x,p)=d(y,p). We consider the problem: Which subspaces of the real line R have the UMP. We prove theorems which imply the following: 1. and J be separated intervals. Then, the sum I∪J has the UMP if and only if at least one of I and J is not compact. m of an odd number of disjoint closed intervals has the UMP. aces [0,1]∪Z and [0,1]∪Q do not have the UMP. be the sum of at most countably many subspaces Xn of R. If each Xn is either an interval or totally disconnected and if at least one of Xn is a noncompact interval, then X has the UMP.