The unique midpoint property of a subspace of the real line

✍ Scribed by Haruto Ohta; Jin Ono

• Book ID: 104295609
• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 113 KB
• Volume: 104
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(99)00026-7

✦ Synopsis

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.