Abstract
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. Let I 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. 2. The sum of an odd number of disjoint closed intervals has the UMP. 3. The spaces [0,1]∪ Z and [0,1]∪ Q do not have the UMP. 4. Let X be the sum of at most countably many subspaces X n of R . If each X n is either an interval or totally disconnected and if at least one of X n is a noncompact interval, then X has the UMP.
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