Vol. 20, No. 1, 1967

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ISSN: 0030-8730
A modification of Morita’s characterization of dimension

Jerry Eugene Vaughan

Vol. 20 (1967), No. 1, 189–196
Abstract

Morita’s characterization of dimension may be stated in the following form. Let R be a metric space. A necessary and sufficient condition that dimR n is that there exists a σ-locally finite base 𝒢 for the topology of R such that dim(G G) n 1 for all G in 𝒢.

The main result of this paper is the following:

THEOREM. Let R be a metric space. A necessary and sufficient condition that dimR n is that there exists a σclosure-preserving base 𝒢 for the topology of R such that dim(G G) n 1 for all G in 𝒢.

Thus the “locally finite” condition in Morita’s characterization can be replaced by the weaker “closure-preserving” condition. A further result is that the “closure-preserving” condition can be replaced by the still weaker condition of “linearly-closure-preserving” provided the “base” condition is strengthened to a “star-base” condition.

Finally, several examples are given which show that the “linearly-closure-preserving” condition is weaker than the “closure -preserving” condition in important ways. In particular, the following is proved.

Theorem. There exists a nonmetric, regular T1-space which has a σ-linearly-closure-preserving star-base.

If the word “linearly” is deleted from the above theorem, the resulting statement is false since Bing has proved that a regular T1-space with a σ-closure-preserving star-base is metrizable.

Mathematical Subject Classification
Primary: 54.70
Milestones
Received: 15 December 1964
Published: 1 January 1967
Authors
Jerry Eugene Vaughan