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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Dimension zero vs measure zero


Author: Ondrej Zindulka
Journal: Proc. Amer. Math. Soc. 128 (2000), 1769-1778
MSC (1991): Primary 28C15, 54F45; Secondary 03E50
Published electronically: September 30, 1999
MathSciNet review: 1646213
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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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Additional Information

Ondrej Zindulka
Affiliation: Department of Mathematics, Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 160 00 Prague 6, Czech Republic
Email: zindulka@mat.fsv.cvut.cz

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05225-9
PII: S 0002-9939(99)05225-9
Keywords: Universal measure zero, topological dimension, zero--dimensional, perfectly meager
Received by editor(s): May 17, 1998
Received by editor(s) in revised form: July 24, 1998
Published electronically: September 30, 1999
Communicated by: Alan Dow
Article copyright: © Copyright 2000 American Mathematical Society