Katetov’s problem

Larson, Paul; Todorcevic, Stevo

doi:10.1090/S0002-9947-01-02936-1

Katetov’s problem
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by Paul Larson and Stevo Todorcevic PDF

Trans. Amer. Math. Soc. 354 (2002), 1783-1791 Request permission

Abstract:

In 1948 Miroslav Katětov 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.

References

Ilijas Farah, OCA and towers in ${\scr P}(\textbf {N})/\textrm {fin}$, Comment. Math. Univ. Carolin. 37 (1996), no. 4, 861–866. MR 1440716
D. H. Fremlin, Consequences of Martin’s axiom, Cambridge Tracts in Mathematics, vol. 84, Cambridge University Press, Cambridge, 1984. MR 780933, DOI 10.1017/CBO9780511896972
Steven A. Gaal, Point set topology, Pure and Applied Mathematics, Vol. XVI, Academic Press, New York-London, 1964. MR 0171253
G. Gruenhage and P. J. Nyikos, Normality in $X^2$ for compact $X$, Trans. Amer. Math. Soc. 340 (1993), no. 2, 563–586. MR 1162102, DOI 10.1090/S0002-9947-1993-1162102-5
R. W. Heath, Screenability, pointwise paracompactness, and metrization of Moore spaces, Canadian J. Math. 16 (1964), 763–770. MR 166760, DOI 10.4153/CJM-1964-073-3
J. Hirschorn, Cohen and random reals, Ph.D. Thesis, University of Toronto, 2000
R. B. Jensen and 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 0289291
F.B. Jones, Concerning normal and completely normal spaces, Bull. Amer. Math. Soc. 43 (1937), 671-677
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1983. An introduction to independence proofs; Reprint of the 1980 original. MR 756630
Kenneth Kunen and Franklin D. Tall, Between Martin’s axiom and Souslin’s hypothesis, Fund. Math. 102 (1979), no. 3, 173–181. MR 532951, DOI 10.4064/fm-102-3-173-181
Paul Larson, An $\textbf {S}_\textrm {max}$ variation for one Souslin tree, J. Symbolic Logic 64 (1999), no. 1, 81–98. MR 1683897, DOI 10.2307/2586753
P. Larson, S. Todorčević, Chain conditions in maximal models, Fund. Math. 168 (2001), no. 1, 77-104
Richard Laver, Random reals and Souslin trees, Proc. Amer. Math. Soc. 100 (1987), no. 3, 531–534. MR 891159, DOI 10.1090/S0002-9939-1987-0891159-4
Tadatoshi Miyamoto, $\omega _1$-Souslin trees under countable support iterations, Fund. Math. 142 (1993), no. 3, 257–261. MR 1220552, DOI 10.4064/fm-142-3-257-261
T. Miyamoto, Iterating semiproper preorders, J. Symbolic Logic, to appear
J. Moore, Ramsey theory on sets of reals, Ph. D. Thesis, University of Toronto, 2000
J. Moore, A counterexample to Katětov’s problem, preprint, October 2000
Teodor C. Przymusiński, Products of normal spaces, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 781–826. MR 776637
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 535835, DOI 10.4064/fm-103-1-47-60
Saharon Shelah and Jindřich Zapletal, Canonical models for $\aleph _1$-combinatorics, Ann. Pure Appl. Logic 98 (1999), no. 1-3, 217–259. MR 1696852, DOI 10.1016/S0168-0072(98)00022-0
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Franklin D. Tall, Normality versus collectionwise normality, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 685–732. MR 776634
Stevo Todorčević, Partition problems in topology, Contemporary Mathematics, vol. 84, American Mathematical Society, Providence, RI, 1989. MR 980949, DOI 10.1090/conm/084
Stevo Todorcevic, Random set-mappings and separability of compacta, Proceedings of the International Conference on Set-theoretic Topology and its Applications (Matsuyama, 1994), 1996, pp. 265–274. MR 1425943, DOI 10.1016/S0166-8641(96)00060-0
Stevo Todorcevic, Chain-condition methods in topology, Topology Appl. 101 (2000), no. 1, 45–82. MR 1730899, DOI 10.1016/S0166-8641(98)00112-6
W. Hugh Woodin, The axiom of determinacy, forcing axioms, and the nonstationary ideal, De Gruyter Series in Logic and its Applications, vol. 1, Walter de Gruyter & Co., Berlin, 1999. MR 1713438, DOI 10.1515/9783110804737