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Katetov's problem


Authors: Paul Larson and Stevo Todorcevic
Journal: Trans. Amer. Math. Soc. 354 (2002), 1783-1791
MSC (2000): Primary 54E35; Secondary 03E35, 03E65, 54E45
DOI: https://doi.org/10.1090/S0002-9947-01-02936-1
Published electronically: November 21, 2001
MathSciNet review: 1881016
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Abstract: In 1948 Miroslav Katetov showed that if the cube $X^{3}$ of a compact space $X$ satisfies the separation axiom T$_{5}$ then $X$ must be metrizable. He asked whether $X^{3}$ can be replaced by $X^{2}$ in this metrization result. In this note we prove the consistency of this implication.


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  • 1. I. Farah, OCA and towers in $\mathcal{P}(\mathbb{N} )/Fin$, Comment. Math. Univ. Carolin. 37 (1996), no. 4, 861-866 MR 98f:03043
  • 2. D.H. Fremlin, Consequences of Martin's axiom, Cambridge Tracts in Mathematics, 84. Cambridge University Press, Cambridge-New York, 1984 MR 86i:03001
  • 3. S.A. Gaal, Point set topology, Pure and Applied Mathematics, Vol. XVI Academic Press, New York-London 1964 MR 30:1484
  • 4. G. Gruenhage, P.J. Nyikos, Normality in $X^{2}$ for compact $X$, Trans. Amer. Math. Soc. 340 (1993), no. 2, 563-586 MR 94b:54009
  • 5. R.W. Heath, Screenability, pointwise paracompactness and metrization of Moore spaces, Canadian J. Math. 16 (1964), 763-770 MR 29:4033
  • 6. J. Hirschorn, Cohen and random reals, Ph.D. Thesis, University of Toronto, 2000
  • 7. R.B. Jensen, R.M. Solovay, Some applications of almost disjoint sets, Mathematical Logic and Foundations of Set Theory (Proc. Internat. Colloq., Jerusalem, 1968) North-Holland, Amsterdam (1970) pp. 84-104 MR 44:6482
  • 8. F.B. Jones, Concerning normal and completely normal spaces, Bull. Amer. Math. Soc. 43 (1937), 671-677
  • 9. M. Katetov, Complete normality of Cartesian products, Fund. Math. 35, (1948) 271-274 MR 10:315h
  • 10. K. Kunen, Set Theory. An introduction to independence proofs, Reprint of the 1980 original. Studies in Logic and the Foundations of Mathematics, 102. North-Holland Publishing Co., Amsterdam-New York, 1983 MR 85e:03003
  • 11. K. Kunen, F.D. Tall, Between Martin's axiom and Souslin's hypothesis, Fund. Math. 102 (1979), no. 3, 173-181 MR 83e:03078
  • 12. P. Larson, An $\mathbb{S} _{max}$ variation for one Souslin tree, J. Symbolic Logic 64 (1999), 81-98 MR 2000g:03118
  • 13. P. Larson, S. Todorcevic, Chain conditions in maximal models, Fund. Math. 168 (2001), no. 1, 77-104
  • 14. R. Laver, Random reals and Souslin trees, Proc. Amer. Math. Soc. 100 (1987), no. 3, 531-534 MR 88g:03068
  • 15. T. Miyamoto, $\omega_{1}$-Souslin trees under countable support iterations, Fund. Math. 142 (1993), 257-261 MR 94f:03060
  • 16. T. Miyamoto, Iterating semiproper preorders, J. Symbolic Logic, to appear
  • 17. J. Moore, Ramsey theory on sets of reals, Ph. D. Thesis, University of Toronto, 2000
  • 18. J. Moore, A counterexample to Katetov's problem, preprint, October 2000
  • 19. T.C. Przymusinski, Products of normal spaces, Handbook of set-theoretic topology, 781-826, North-Holland, Amsterdam-New York, 1984 MR 86c:54007
  • 20. J. Roitman, Adding a random or a Cohen real: topological consequences and the effect on Martin's axiom, Fund. Math. 103 (1979), no. 1, 47-60 MR 81h:03098
  • 21. S. Shelah, J. Zapletal, Canonical Models for $\aleph_{1}$ Combinatorics, Annals of Pure and Applied Logic 98 (1999), 217-259 MR 2000m:03113
  • 22. V.E. Sneider, V. E. Continuous images of Suslin and Borel sets. Metrization theorems, (Russian) Doklady Akad. Nauk SSSR (N.S.) 50, (1945). 77-79 MR 14:782d
  • 23. F. Tall, Normality versus collectionwise normality, Handbook of set-theoretic topology, 685-732, North-Holland, Amsterdam-New York, 1984 MR 86m:54022
  • 24. S. Todorcevic, Partition problems in topology, Contemporary Mathematics, 84. American Mathematical Society, Providence, RI, 1989 MR 90d:04001
  • 25. S. Todorcevic, Random set-mappings and separability of compacta, Proceedings of the International Conference on Topology and its Applications (Matsuyama, 1994). Topology Appl. 74 (1996), no. 1-3, 265-274 MR 97j:03099
  • 26. S. Todorcevic, Chain-condition methods in topology, Topology Appl. 101 (2000), no. 1, 45-82 MR 2001a:54055
  • 27. W.H. Woodin, The axiom of determinacy, forcing axioms, and the nonstationary ideal, DeGruyter Series in Logic and Its Applications, vol. 1, 1999 MR 2001e:03001

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Additional Information

Paul Larson
Affiliation: Department of Mathematics, University of Toronto, Toronto M5S 3G3, Canada
Email: larson@math.toronto.edu

Stevo Todorcevic
Affiliation: C.N.R.S. (7056), Université Paris VII, 75251 Paris Cedex 05, France
Email: stevo@math.jussieu.fr

DOI: https://doi.org/10.1090/S0002-9947-01-02936-1
Keywords: Compactness, metrizability, T$_{5}$, forcing
Received by editor(s): November 27, 2000
Received by editor(s) in revised form: July 30, 2001
Published electronically: November 21, 2001
Additional Notes: This work was done while the authors were in residence at the Mittag-Leffler Institute. We thank the Institute for its hospitality.
Article copyright: © Copyright 2001 American Mathematical Society

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