• Abstract
• Introduction
• NEXT: Proof of the theorems
• An example and remarks
• References

Introduction

Every subspace of the real line with the Euclidean metric d has the following two properties: (a) There exist no four distinct points and such that for each i = 1, 2; (b) There exist no four distinct points and such that . The first purpose of this note is to prove conversely the following theorem:

The second purpose is to show that a separable metric space (X, d) having (a) alone need not be homeomorphic to a subspace of . Since the circle with a metric inherited from the Euclidean metric d on the plane has (b), a metric space (X, d) having (b) alone need not be homeomorphic to a subspace of .

In the remaining part of this section, we discuss the relationship between the property (a) and other metric properties. Recall from [1] that a metric space (X, d) has the unique midset property (abbreviated UMP) if for every pair of distinct points x and y, there exists one and only one point p such that d(x, p) = d(y, p). A metric space (X, d) having UMP has (a) and the converse holds if (X, d) is connected (cf. [7], Lemma 2.1). Berard [1] showed that a connected metric space consisting of more than one point and having UMP is homeomorphic to an interval in . Nadler [7] showed that a locally compact and separable metric space having UMP is homeomorphic to a subspace of . However, UMP is too restrictive to describe a subspace of , because not all subspaces X of have a compatible metric d for which (X, d) has UMP (see Remarks 2). The following theorem can be proved by the same way as the proof of Nadler's theorem stated above.

An alternative proof of Theorem 2 will be given later.
Two subsets A and B of a metric space (X, d) are said to be congruent if there exists a bijection such that d(x, y) = d(f(x), f(y)) whenever x and y in A. We say that a metric space has the unique triangle property (abbreviated UTP) if no distinct subsets of cardinality 3 are congruent. Janos [4] proved that the dimension of a locally compact and separable metric space having UTP is at most 1. A metric space (X, d) having UTP has the property (a), because if for each i = 1, 2, then the sets and are congruent. Hence, Theorem 2 sharpens Janos's theorem. For further background, the reader is referred to [3] and [5].

For a point p of a metric space (X, d) and , we write and . Moreover, we often write simply X to denote a metric space (X, d) and abbreviate ^^ closed-and-open' to ^^ clopen'. The letter denotes the set of positive integers. Other terms and notation follow [2].