Dimension zero vs measure zero

Zindulka, Ondřej

doi:10.1090/S0002-9939-99-05225-9

Dimension zero vs measure zero
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by Ondřej Zindulka PDF

Proc. Amer. Math. Soc. 128 (2000), 1769-1778 Request permission

Abstract:

The following problem is discussed: If $X$ is a topological space of universal measure zero, does it have also dimension zero? It is shown that in a model of set theory it is so for separable metric spaces and that under the Martin’s Axiom there are separable metric spaces of positive dimension yet of universal measure zero. It is also shown that for each finite measure in a metric space there is a zero–dimensional subspace that has full measure. Similar questions concerning perfectly meager sets and other types of small sets are also discussed.

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