Vol. 84, No. 2, 1979

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ISSN: 0030-8730
A characterization of dimension of topological spaces by totally bounded pseudometrics

Jeroen Bruijning

Vol. 84 (1979), No. 2, 283–289
Abstract

For a compact metrizable space X, for a metric d on X, and for 𝜖 > 0, the number N(𝜖,X,d) is defined as the minimum number of sets of d-diameter not exceeding 𝜖 required to cover X. A classical theorem characterizes the topological dimension of X in terms of the numbers N(𝜖,X,d). In this paper, two extensions of this result are given: (i) a direct one, to separable metrizable spaces, involving totally bounded metrics; (ii) a more complicated one, involving the set of continuous totally bounded pseudometrics on the space as well as a special order on this set.

Mathematical Subject Classification 2000
Primary: 54F45
Milestones
Received: 7 June 1978
Revised: 20 February 1979
Published: 1 October 1979
Authors
Jeroen Bruijning