Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-20T12:29:03.337Z Has data issue: false hasContentIssue false
  • English
  • Français

The Character of Certain Closed Sets

Published online by Cambridge University Press:  20 November 2018

Mary Anne Swardson
Mary Anne Swardson*
Affiliation:
Ohio University, Athens, Ohio
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X be a topological space and let A ⊂ X. The character of A in X is the minimal cardinal of a base for the neighborhoods of A in X. Previous studies have shown that the character of certain subsets of X (or of X2) is related to compactness conditions on X. For example, in [12], Ginsburg proved that if the diagonal

of a space X has countable character in X2, then X is metrizable and the set of nonisolated points of X is compact. In [2], Aull showed that if every closed subset of X has countable character, then the set of nonisolated points of X is countably compact. In [18], we noted that if every closed subset of X has countable character, then MA + ┐ CH (Martin's axiom with the negation of the continuum hypothesis) implies that X is paracompact.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 1 , 01 February 1984 , pp. 38 - 57
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Alexandroff, P. and Urysohn, P., On compact topological spaces, Trudy Mat. Inst. Steklov 31, 95 (1950), (Russian).Google Scholar
2. Aull, C. E., Closed set countability axioms, Indag. Math. 28 (1966), 311–316.Google Scholar
3. Aull, C. E., A generalization of a theorem of Aquaro, Bull. Austral. Math. Soc. 9 (1973), 105–108.Google Scholar
4. Blair, R. L., Spaces in whicn special sets are z-embedded, Can. J. Math. 28 (1976), 673–690.Google Scholar
5. Chaber, J., Conditions which imply compactness in count ably compact spaces, Bull. Acad. Pol Sci. Math. 24 (1976), 993–998.Google Scholar
6. Comfort, W. W. and Negrepontis, S., The theory of ultra]liters, Die Grundlehren der Math. Wissenschaften 211 (Springer-Verlag, New York-Heidelburg-Berlin, 1974).Google Scholar
7. van Douwen, E. K., Transfer of information about βN-N via open remainder maps, Illinois J. Math., to appear.Google Scholar
8. van Douwen, E. K., Remote points, Dissertationes Math. 188, PWN, Warsaw (1980).Google Scholar
9. Engelking, R., General topology (PWN, Warsaw, 1975); English trans. (PWN, Warsaw, 1977).Google Scholar
10. Fine, N. J. and Gillman, L., Extension of continuous functions in βN, Bull. Amer. Math. Soc 66 (I960), 376–381.Google Scholar
11. Gillman, L. and Jerison, M., Rings of continuous functions, University Series in Higher Math. (Van Nostrand, Princeton, 1960).CrossRefGoogle Scholar
12. Ginsburg, J., The metrizability of spaces whose diagonals have a countable base, Can. Math. Bull 20 (1977), 513–514.Google Scholar
13. Kennison, J. F., m-pseudocompactness, Trans. Amer. Math. Soc. 104 (1962), 436–442.Google Scholar
14. Moran, W., Measures on metacompact spaces, Proc. London Math. Soc. (3) 20 (1970), 507–524.Google Scholar
15. Ostaszewski, A. J., On countably compact, perfectly normal spaces, J. London Math. Soc. 74 (1976), 505–516.Google Scholar
16. Rudd, D., A note on zero-sets in the Stone-Čech compactification, Bull. Austral. Math. Soc. 12 (1975), 227–230.Google Scholar
17. Steen, L. A. and Seeback, J. A. Jr., Counterexamples in topology, second edition (Springer-Verlag, New York-Heidelburg-Berlin, 1978).CrossRefGoogle Scholar
18. Swardson, M. A., A note on the closed character of a topological space, Topology Proc. 4 (1979), 601–608.Google Scholar
19. Swardson, M. A., A generalization of F-spaces and some topological characterizations of GCII, Trans. Amer Math. Soc. 279 (1983), 661–675.Google Scholar
20. Terada, T., On spaces whose Stone-Cech compactification is Oz, Pacific J. Math. 85 (1979), 231–237.Google Scholar
21. Wage, M. L., Countable paracompactness, normality, and Moore spaces, Proc. Amer. Math. Soc 57 (1976), 183–188.Google Scholar
22. Weiss, W., Countably compact spaces and Martin's axiom, Can. J. Math. 30 (1978), 243–249.Google Scholar