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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 σ− closure-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.
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