Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Convergent free sequences in compact spaces

Authors: I. Juhász and Z. Szentmiklóssy
Journal: Proc. Amer. Math. Soc. 116 (1992), 1153-1160
MSC: Primary 54D35
MathSciNet review: 1137223
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that if $ \kappa $ is an uncountable regular cardinal and a compact $ {T_2}$ space $ X$ contains a free sequence of length $ \kappa $, then $ X$ also contains such a sequence that is convergent. This implies that under $ {\text{CH}}$ every nonfirst countable compact $ {T_2}$ space contains a convergent $ {\omega _1}$-sequence and every compact $ {T_2}$ space with a small diagonal is metrizable, thus answering old questions raised by the first author and M. Hušek, respectively.

References [Enhancements On Off] (What's this?)

  • [BSV] B. Balcár, P. Simon, and P. Vojtáš, Refinement properties and extensions of filters in Boolean algebras, Trans. Amer. Math. Soc. 267 (1981), 265-283. MR 621987 (82k:06014)
  • [D1] A. Dow, Compact spaces of countable tightness in the Cohen model, Set Theory and its Appl. (J. Steprāns and S. Watson, eds.), Lecture Notes in Math., vol. 1401, Springer-Verlag, New York, 1989, pp. 55-67. MR 1031765 (91a:54004)
  • [D2] -, An introduction to applications of elementary submodels in topology, Topology Proc. 13 (1988), 17-72. MR 1031969 (91a:54003)
  • [E] R. Engelking, General topology, Helderman Verlag, Berlin, 1989. MR 1039321 (91c:54001)
  • [H] M. Hušek, Topological spaces without $ \kappa $-accessible diagonal, Comment. Math. Univ. Carolin. 18 (1977), 777-788. MR 0515009 (58:24198)
  • [J1] I. Juhász, Cardinal function-ten years later, Math. Centre Tract, vol. 123, North-Holland, Amsterdam, 1980.
  • [J2] -, A weakening of club, with applications to topology, Comment. Math. Univ. Carolin. 29 (1988), 767-773. MR 982796 (90d:54005)
  • [J3] -, On the minimum character of points in compact space, Proc. 1989 Topology Conf. Pécs (to appear).
  • [J4] -, Two set-theoretic problems in topology, Proc. Fourth Prague Sympos. on Gen. Topology Part A, Springer-Verlag, New York, 1977, pp. 115-123. MR 0458350 (56:16553)
  • [vM] J. van Mill, An introduction to $ \beta \omega $, Handbook of Set-Theoretic Topology, 1984, pp. 503-567. MR 776630 (86f:54027)
  • [O] A. Ostaszewski, On countably compact, perfectly normal spaces, J. London Math. Soc. (2) 14 (1976), 505-516. MR 0438292 (55:11210)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54D35

Retrieve articles in all journals with MSC: 54D35

Additional Information

Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society