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The Number of Closed Subsets of a Topological Space

Published online by Cambridge University Press:  20 November 2018

R. E. Hodel
R. E. Hodel*
Affiliation:
Duke University, Durham, North Carolina 27706
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Let X be an infinite topological space, let 𝔫 be an infinite cardinal number with 𝔫 ≦ |X|. The basic problem in this paper is to find the number of closed sets in X of cardinality 𝔫. A complete answer to this question for the class of metrizable spaces has been given by A. H. Stone in [31], where he proves the following result. Let X be an infinite metrizable space of weight 𝔪, let 𝔫 ≦ |X|.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 2 , 01 April 1978 , pp. 301 - 314
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Arhangel'skiï, A. V., On bicompacta hereditarily satisfying Suslins condition. Tightness and free sequences, Soviet Math. Dokl. 12 (1971), 1253–1256.Google Scholar
2. Boone, J., On irreducible spaces, Bull. Austral. Math. Soc. 12 (1975), 143–148.Google Scholar
3. Burke, D. K., On p-spaces and wA-spaces, Pacific J. Math. 35 (1970), 285–296.Google Scholar
4. Burke, D. K. and Hodel, R. E., The number of compact subsets of a topological space, Proc. Amer. Math. Soc. 58 (1976), 363–368.Google Scholar
5. Comfort, W. W. and Hager, A. W., Estimates for the number of real-valued continuous functions, Trans. Amer. Math. Soc. 150 (1970), 619–631.Google Scholar
6. van Douwen, E. K., Functions from the integers to the integers and topology, to appear.Google Scholar
7. Engelking, R., General topology (Warszawa, 1977).Google Scholar
8. Fedorcuk, V. V., On the cardinality of hereditarily separable compact Hausdorff spaces, Soviet Math. Dokl. 16 (1975), 651–655.Google Scholar
9. Gillman, L. and Jerison, M., Rings of continuous functions (Van Nostrand, Princeton, 1960).Google Scholar
10. de Groot, J., Discrete subspaces of Hausdorff spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 537–544.Google Scholar
11. Hajnal, A. and Juhâsz, I., Discrete subspaces of topological spaces, Indag. Math. 29 (1967), 343–356.Google Scholar
12. Hajnal, A. and Juhâsz, I. Discrete subspaces of topological spaces, II, Indag. Math. 31 (1969), 18–30.Google Scholar
13. Hajnal, A. and Juhâsz, I. On hereditarily a-Lindelof and hereditarily a-separable spaces, Ann. Univ. Sci. Budapest Eôtvôs Sect. Math. 11 (1968), 115–124.Google Scholar
14. Hajnal, A. and Juhâsz, I. Some remarks on a property of topological cardinal functions, Acta Math. Acad. Sci. Hung. 20 (1969), 25–37.Google Scholar
15. Hajnal, A. and Juhâsz, I. On the number of open sets, Ann. Univ. Sci. Budapest, Sect. Math. 16 (1973), 99–102.Google Scholar
16. Hajnal, A. and Juhâsz, I. A consistency result concerning hereditarily a-Lindelof spaces, Acta Math. Acad. Sci. Hung. 24 (1973), 307–312.Google Scholar
17. Juhâsz, I., Cardinal functions in topology, Math. Centre Tract 34, Amsterdam, 1971.Google Scholar
18. Juhâsz, I. Remarks on cardinal functions, Colloquia Mathematica Societatis Janos Bolyai 8, Topics in Topology, Keszthely (Hungary), 1972, 449–450.Google Scholar
19. Juhâsz, I., Nagy, Zs., and Weiss, W., On countably compact, locally countable spaces, Preprint of the Mathematical Institute of the Hungarian Academy of Sciences, No. 7, 1976.Google Scholar
20. Krivorucko, A. I., On the cardinality of the set of continuous functions, Soviet Math. Dokl. 13 (1972), 1364–1367.Google Scholar
21. Krivorucko, A. I., The cardinality and density of spaces of mappings, Soviet Math. Dokl. 16 (1975), 281–285.Google Scholar
22. Moore, R. L., Foundations of point set theory, Amer. Math. Soc. Colloquium Publ. 13, rev. ed. 1962.Google Scholar
23. Nagata, J. and Siwiec, F., A note on nets and metrization, Proc. Japan Acad. 44 (1968), 623–627.Google Scholar
24. Novak, J., On the Cartesian product of two compact spaces, Fund. Math. 1+0 (1953), 106–112.Google Scholar
25. Okuyama, A., Some generalizations of metric spaces, their metrization theorems and product theorems, Sci. Rep. Tokyo Kyoiku Daigaku, Ser. A 9 (1967), 236–254.Google Scholar
26. Rudin, W., Homogeneity problems in the theory of Cech compactifications, Duke Math. J. 23 (1955), 409–419.Google Scholar
27. Sapirovskiï, B., On discrete subspaces of topological spaces; weight, tightness and Suslin number, Soviet Math. Dokl. 13 (1972), 215–219.Google Scholar
28. Sapirovskiï, B. Canonical sets and character. Density and weight in compact spaces, Soviet Math. Dokl. 15 (1974), 1282–1287.Google Scholar
29. Smith, J. C., A remark on irreducible spaces, Proc. Amer. Math. Soc. 57 (1976), 133–138.Google Scholar
30. Smith, J. C., Characterizations of generalized paracompact spaces, Colloq. Math. 35 (1976), 29–39.Google Scholar
31. Stone, A. H., Cardinals of closed sets, Mathematika 6 (1959), 99–107.Google Scholar
32. Stone, A. H., Non-separable Borel sets, Rozprawy Matematyczne 28 (1962), 1–40.Google Scholar
33. Tarski, A., Sur la decomposition des ensembles en sous-ensembles presque disjoints, Fund. Math. 12 (1928), 188–205.Google Scholar
34. Vaughan, J., Discrete sequences of points, to appear.Google Scholar
35. Wicke, H. H. and Worrell, J. M. Jr., Characterizations of developable spaces, Can. J. Math. 17 (1965), 820–830.Google Scholar
36. Willard, S., General topology (Addison-Wesley, Reading, 1968).Google Scholar