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On Metrizability of Topological Spaces

Published online by Cambridge University Press:  20 November 2018

Carlos J. R. Borges
Carlos J. R. Borges*
Affiliation:
University of California, Davis, California
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Our present work is divided into three sections. In §2 we study the metrizability of spaces with a Gδ-diagonal (see Definition 2.1). In §3 we study the metrization of topological spaces by means of collections of (not necessarily continuous) real-valued functions on a topological space. Our efforts, in §§2 and 3, are directed toward answering the following question: “Is every normal, metacompact (see Definition 2.4) Moore space a metrizable space?” which still remains unsolved. (However, Theorems 2.12 through 2.15 and Theorem 3.1 may be helpful in answering the preceding question.) In §4 we prove an apparently new necessary and sufficient condition for the metrizability of the Stone-Čech compactification of a metrizable space and hence for the compactness of a metric space.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 795 - 804
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

This research was supported by NSF Grant GP-4770.

References

1. Bing, R. H., Metrization of topological spaces, Can. J. Math. 3 (1951), 175–186.Google Scholar
2. Borges, C. J. R., On stratifiable spaces, Pacific J. Math. 17 (1966), 1–16.Google Scholar
3. Ceder, J. G., Some generalizations of metric spaces, Pacific J. Math. 11 (1961), 105–125.Google Scholar
4. Corson, H. and Michael, E., Metrizability of certain countable unions, Illinois J. Math. 8 1964), 351–360.Google Scholar
5. Gillman, L. and Gerison, M., Rings of continuous functions (Van Nostrand, Princeton, N.J., 1960).10.1007/978-1-4615-7819-2CrossRefGoogle Scholar
6. Henriksen, M. and Isbell, J., Some properties of compactifications, Duke Math. J. 25 (1958), 83–105.Google Scholar
7. Heath, R. W., Screenability, pointwise paracompactness, and metrization of Moore spaces, Can. J. Math. 26 (1964), 763–770.Google Scholar
8. Kelley, J. K., General topology (Van Nostrand, Princeton, N.J., 1955).Google Scholar
9. Morita, K., Products of normal spaces with metric spaces, Math. Ann. 154 (1964), 365–382.Google Scholar
10. Okuyama, A., On metrizability of M-spaces, Proc. Japan Acad. 40 (1964), 176–179.Google Scholar
11. Proizvolov, V., One-to-one mappings onto metric spaces, Soviet Math. Dokl. 5 (1964), 1321–1322.Google Scholar
12. Smirnov, Yu. M., On metrization of topological spaces, Amer. Math. Soc. Transi. 8 (Ser. 1), 62–67.Google Scholar
13. Stone, A. H., Metrizability of unions of spaces, Proc. Amer. Math. Soc. 7 (1956), 690–700.Google Scholar
14. Traylor, D. R., Concerning metrizability of pointwise paracompact Moore spaces, Can. J. Math. 16 (1964), 407–411.Google Scholar