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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
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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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  • 1. James E. Baumgartner and Richard Laver, Iterated perfect-set forcing, Ann. Math. Logic 17 (1979), no. 3, 271–288. MR 556894, https://doi.org/10.1016/0003-4843(79)90010-X
  • 2. Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR 776622
  • 3. Ryszard Engelking, Teoria wymiaru, Państwowe Wydawnictwo Naukowe, Warsaw, 1977 (Polish). Biblioteka Matematyczna, Tom 51. [Mathematics Library, Vol. 51]. MR 0482696
    Ryszard Engelking, Dimension theory, North-Holland Publishing Co., Amsterdam-Oxford-New York; PWN—Polish Scientific Publishers, Warsaw, 1978. Translated from the Polish and revised by the author; North-Holland Mathematical Library, 19. MR 0482697
  • 4. D. H. Fremlin, Consequences of Martin’s axiom, Cambridge Tracts in Mathematics, vol. 84, Cambridge University Press, Cambridge, 1984. MR 780933
  • 5. W. Hurewicz, Une remarque sur l'hypothese du continu, Fund. Math. 19 (1932), 8-9.
  • 6. Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
  • 7. Arnold W. Miller, Mapping a set of reals onto the reals, J. Symbolic Logic 48 (1983), no. 3, 575–584. MR 716618, https://doi.org/10.2307/2273449
  • 8. Arnold W. Miller, Special subsets of the real line, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 201–233. MR 776624
  • 9. Jacques Stern, Partitions of the real line into ℵ₁ closed sets, Higher set theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1977), Lecture Notes in Math., vol. 669, Springer, Berlin, 1978, pp. 455–460. MR 520200
  • 10. J. E. Vaughan, Small uncountable cardinals and topology, Open problems in topology, North Holland, 1991. CMP 91:03

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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: 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