Dimension zero vs measure zero

Author:
Ondrej Zindulka

Journal:
Proc. Amer. Math. Soc. **128** (2000), 1769-1778

MSC (1991):
Primary 28C15, 54F45; Secondary 03E50

DOI:
https://doi.org/10.1090/S0002-9939-99-05225-9

Published electronically:
September 30, 1999

MathSciNet review:
1646213

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The following problem is discussed: If 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.

**1.**J. Baumgartner and R. Laver,*Iterated perfect set forcing*, Ann. Math. Logic**17**(1979), 271-288. MR**81a:03050****2.**E. K. van Douwen,*The integers and topology*, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, Amsterdam, New York, Oxford, 1984. MR**87f:54008****3.**R. Engelking,*Dimension theory*, North Holland, Amsterdam, 1978. MR**58:2753b****4.**D. H. Fremlin,*Consequences of Martin's Axiom*, Cambridge University Press, Cambridge, 1984. MR**86i:03001****5.**W. Hurewicz,*Une remarque sur l'hypothese du continu*, Fund. Math.**19**(1932), 8-9.**6.**K. Kunen,*Set theory. An Introduction to Independence Proofs*, North-Holland, Amsterdam, 1980. MR**82f:03001****7.**A. W. Miller,*Mapping a set of reals onto the reals*, J. Symbolic Logic (1983), no. 48, 575-584. MR**84k:03125****8.**-,*Special subsets of the real line*, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, Amsterdam, New York, Oxford, 1984. MR**86i:54037****9.**J. Stern,*Partitions of the real line into closed sets*, Higher Set Theory (G. H. Müller and D. S. Scott, eds.), Springer Verlag, 1978, LNM 669. MR**80f:03058****10.**J. E. Vaughan,*Small uncountable cardinals and topology*, Open problems in topology, North Holland, 1991. CMP**91:03**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
28C15,
54F45,
03E50

Retrieve articles in all journals with MSC (1991): 28C15, 54F45, 03E50

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:
https://doi.org/10.1090/S0002-9939-99-05225-9

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