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Cardinals of closed sets

Published online by Cambridge University Press:  26 February 2010

A. H. Stone
A. H. Stone
Affiliation:
Manchester University.
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Extract

The Čech compactification of the set of integers is known [6] to have the remarkable property that it has no closed subsets of cardinal ℵ0 or c, every infinite closed subset of it having 2c points. The main object of the present paper is to investigate whether similar gaps in the cardinals of closed subsets can occur in metric spaces. We shall see that the situation there is rather different; if the generalized continuum hypothesis is assumed, there are no gaps, and in any case the missing cardinals, if any, must be big rather than small. The main results are obtained in §3; in particular, we completely determine the cardinals of the closed subsets of complete metric spaces, and also how many closed subsets of each cardinal there are. The methods depend on a study of the discrete subsets of metric spaces, which is carried out in §2, and which may be of independent interest. In conclusion, we briefly consider some fragmentary results for non-metric spaces, in §4. Throughout, we assume the axiom of choice but not the continuum hypothesis.

Type
Research Article
Information
Mathematika , Volume 6 , Issue 2 , December 1959 , pp. 99 - 107
Copyright
Copyright © University College London 1959

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References

1.Alexandroff, P. and Urysohn, P., Mémoire sur les espaces topologiques compacts, Verh. Kon. Ned. Akad. v. Wetens., 14 (Amsterdam, 1929).Google Scholar
2.Bing, R. H., “Metrization of topological spaces”, Canadian J. of Math., 3 (1951), 175–186CrossRefGoogle Scholar
3.Gál, I. S., “On a generalized notion of compactness I, II”, Proc. Kon. Ned. Akad. v. Wetens., 60 (A) (1957), 421–435.Google Scholar
4.Hausdorff, F., “Erweiterung einer Homöomorphie”, Fundamenta Math., 16 (1930), 353–360.CrossRefGoogle Scholar
5.Kuratowski, C., Topologie I (Warsaw, 1948).Google Scholar
6.Novák, J., “On the cartesian product of two compact spaces”, Fundamenta Math., 40 (1953), 107–112.Google Scholar
7.Schmidt, F. K., “Über die Dichte metrischer Raüme”, Math. Annalen, 106 (1932), 457–472.CrossRefGoogle Scholar
8.Sierpński, W., Cardinal and ordinal numbers (Warsaw, 1958).Google Scholar
9.Stone, A. H., “Paracompactness and product spaces”, Bull. American Math. Soc., 54 (1948), 977–982.CrossRefGoogle Scholar