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Uncountable Discrete Sets in Extensions and Metrizability

Published online by Cambridge University Press:  20 November 2018

Murray Bell  and
John Ginsburg
Murray Bell
Affiliation:
University of Manitoba, Winnipeg Manitoba, CanadaR3T 2N2
John Ginsburg
Affiliation:
University of Winnipeg, Winnipeg Manitoba, CanadaR3B 2E9
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Abstract

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If X is a topological space then exp X denotes the space of non-empty closed subsets of X with the Vietoris topology and λX denotes the superextension of X Using Martin's axiom together with the negation of the continuum hypothesis the following is proved: If every discrete subset of exp X is countable the X is compact and metrizable. As a corollary, if λX contains no uncountable discrete subsets then X is compact and metrizable. A similar argument establishes the metrizability of any compact space X whose square X × X contains no uncountable discrete subsets.

Keywords

54A2554E3554B1054B2054D3002K05uncountable discrete setspace of closed setssuperextensionsquaremetrizabilitycompactnessMartin's axiom
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 4 , 01 December 1982 , pp. 472 - 477
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Alexandroff, P. and Urysohn, P., Memoire sur les espaces topologiques compacts, Verh. Wetensch. Afd. Natuurk. Sect. 14, 1 (1929-31), 1-96.Google Scholar
2. Engleking, R., Outline of General Topology, North-Holland Publishing company, Amsterdam 1968.Google Scholar
3. Ginsburg, J., Contributions to the Theory of Hyperspaces, Doctoral dissertation, University of Manitoba, 1975.Google Scholar
4. Juhasz, I., Cardinal Functions in Topology, Math. Centre Tracts 34, Mathematical Centre, Amsterdam 1971.Google Scholar
5. Kunen, K., Products of S-spaces, preprint.Google Scholar
6. Malyhin, V. I., On tightness and Souslin number in exp X and in a product of spaces, Soviet Math. Dokl. Vol. 13 (1972) No. 2.Google Scholar
7. Michael, E., Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-183.Google Scholar
8. Przymusinski, T., Products of perfectly normal spaces, Fund. Math. c.v. III, 1980, 129-136.Google Scholar
9. Rudin, M. E., Lectures on Set Theoretic Topology, CBMS series, No. 23, AMS, 1975.Google Scholar
10. Sneider, V. E., Continuous images of Suslin and Borel sets; Metrization theorems, Dokl. Akad. Nauk. SSSR 50 (1945), 77-79.Google Scholar
11. Solovay, R. M. and Tennenbaum, S., Iterated Cohen extensions and Souslin's problem, Ann. of Math. 94 (1971), 201-245.Google Scholar
12. Szentmiklossy, Z., S-spaces and L-spaces under Martin's axiom, Colloq. Math. Soc. János Bolyai, 23 (1978), 1139-1145.Google Scholar
13. Verbeek, A., Superextensions of topological spaces, Math. Centre Tracts 41, Mathematics Centre, Amsterdam, 1972.Google Scholar
14. Zenor, P., Hereditary m-separability and the hereditary m-Lindelof property in product spaces and function spaces, Fund. Math., c.v. I (1980), 175-180.Google Scholar