Electronic Version 18 (1993), 75-87

Yasuano Hattori and Haruto Ohta

We prove: A separable metrizable space is homeomorphic to a subspace of the real line if and only if it admits a metric which induces the original topology and satisfies

1. the cardinality of any subset consisting of points which are equidistant from two distinct points is at most 1; and

2. the cardinality of any subset consisting of points which are equidistant from a point is at most 2.

A separable metric space satisfying 1 or 2 alone need not be homeomorphic to a subspace of the real line.

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volume 18: table of contents
topology proceedings Electronic Version