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Proceedings of the American Mathematical Society

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

 

 

A regular topological space having no closed subsets of cardinality $ \aleph\sb 2$


Authors: Martin Goldstern, Haim I. Judah and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 111 (1991), 1151-1159
MSC: Primary 54A25; Secondary 03E50, 03E75
DOI: https://doi.org/10.1090/S0002-9939-1991-1052572-9
MathSciNet review: 1052572
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Abstract: Using $ {\diamondsuit _{{\lambda ^ + }}}$, we construct a regular topological space in which all closed sets are of cardinality either $ < \lambda {\text{or}} \geq {{\text{2}}^{{\lambda ^ + }}}$. In particular (answering a question of Juhász) there is always a regular space in which no closed set has cardinality $ {\aleph _2}$.


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  • [vD] E. K. van Douwen, Cardinal functions on compact $ F$-spaces, and weakly countably compact Boolean algebras, Fund. Math. 114, 235-256.
  • [G] John Gregory, Higher Souslin trees and the generalized continuum hypothesis, J. Symbolic Logic 41 (1976), no. 3, 663–671. MR 0485361, https://doi.org/10.2307/2272043
  • [HJ] A. Hajnal and I. Juhász, On hereditarily $ \alpha $-Lindelöf and $ \alpha $-separable spaces II, Fund. Math. 81, 147-158.
  • [J] Kenneth Kunen and Jerry E. Vaughan (eds.), Handbook of set-theoretic topology, North-Holland Publishing Co., Amsterdam, 1984. MR 776619
  • [Ho] R. Hodel, Cardinal functions. I, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 1–61. MR 776620
  • [Hu] M. Hušek, Omitting cardinal functions by topological spaces, General topology and its relations to modern analysis and algebra, V (Prague, 1981) Sigma Ser. Pure Math., vol. 3, Heldermann, Berlin, 1983, pp. 387–394. MR 698427
  • [S] Saharon Shelah, Models with second order properties. III. Omitting types for 𝐿(𝑄), Arch. Math. Logik Grundlag. 21 (1981), no. 1-2, 1–11. MR 625527, https://doi.org/10.1007/BF02011630

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1052572-9
Article copyright: © Copyright 1991 American Mathematical Society