Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T06:53:02.688Z Has data issue: false hasContentIssue false

On Expanding Locally Finite Collections

Published online by Cambridge University Press:  20 November 2018

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.

A space X is in-expandable, where m is an infinite cardinal, if for every locally finite collection {Hα| αA} of subsets of X with |A| ≦ m(cardinality of Am) there exists a locally finite collection of open subsets {Gα| αA} such that HαGα for every αA. X is expandable if it is m-expandable for every cardinal m. The notion of expandability is closely related to that of collection wise normality introduced by Bing [1], X is collectionwise normal if for every discrete collection of subsets {Hα|αA} there is a discrete collection of open subsets {Gα|αA} such that HαGα for every αA. Expandable spaces share many of the properties possessed by collectionwise normal spaces. For example, an expandable developable space is metrizable and an expandable metacompact space is paracompact.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Bing, R. H., Metrization of topological spaces, Can. J. Math. 3 (1951), 175186.Google Scholar
2. Borges, C. J. R., Onmetrizability of topological spaces, Can. J. Math. 20 (1968), 795804.Google Scholar
3. Dieudonné, J., Unegénéralisation des espaces compacts, J. Math. Pures Appl. (9) 23 (1944), 6576.Google Scholar
4. Dowker, C. H., Oncountablyparacompact spaces, Can. J. Math. 3 (1951), 219224.Google Scholar
5. Engelking, R., Outline of general topology (Interscience, New York, 1968).Google Scholar
6. Hanai, S., Inverse images of closed mappings. I, Proc. Japan Acad. 37 (1961), 298301.Google Scholar
7. Hayashi, Y., Oncountablymetacompact spaces, Bull. Univ. Osaka Prefecture Ser.A 8 (1959/60), 161164.Google Scholar
8. Heath, R. W., Screenability, pointwiseparacompactness and metrization of Moore spaces, Can. J. Math. 16 (1964), 663670.Google Scholar
9. Hodel, R. E., Sum theorems for topological spaces, Pacific J. Math. 30 (1969), 5965.Google Scholar
10. Ishii, T., On closed mappings and M-spaces. II, Proc. Japan Acad. 43 (1967), 757761.Google Scholar
11. Ishikawa, F., Oncountablyparacompact spaces, Proc. Japan Acad. 31 (1955), 686687.Google Scholar
12. Katëtov, M., Extension of locally finite coverings, Colloq. Math. 6 (1958), 145151. (Russian)Google Scholar
13. Mack, J., On a class of countablyparacompact spaces, Proc. Amer. Math. Soc. 16 (1965), 467472.Google Scholar
14. Mack, J., Directed covers and paracompact spaces, Can. J. Math. 19 (1967), 649654.Google Scholar
15. Mansfield, M. J., Oncountablyparacompact normal spaces, Can. J. Math. 9 (1957), 443449.Google Scholar
16. McAuley, L. F., A note on complete collectionwise normality and paracompactness, Proc. Amer. Math. Soc. 9 (1958), 796799.Google Scholar
17. Michael, E. A., A note on paracompact spaces, Proc. Amer. Math. Soc. 4 (1953), 831838.Google Scholar
18. Michael, E. A., Point-finite and locally finite coverings, Can. J. Math. 7 (1955), 275279.Google Scholar
19. Morita, K., Products of normal spaces with metric spaces, Math. Ann. 154 (1964), 365382.Google Scholar
20. Morita, K., Paracompactness and product spaces, Fund. Math. 50 (1961/62), 223236.Google Scholar
21. Morita, K., On closed mappings, Proc. Japan Acad. 32 (1956), 539543.Google Scholar
22. Morita, K. and Hanai, S., Closed mappings and metric spaces, Proc. Japan Acad. 32 (1956), 1014.Google Scholar
23. Mrôwka, S., On local topological properties, Bull. Acad. Polon. Sci. CI. III 5 (1957), 951956.Google Scholar
24. Nagami, K., Paracompactness and strong screenability, Nagoya Math. J. 8 (1955), 8388.Google Scholar
25. Okuyama, A., Some generalizations of metric spaces, their metrization theorems and product spaces, Sci. Rep. Tokyo KyoikuDaigaku Sect. A 9 (1967), 6078.Google Scholar
26. Singal, M. K. and Arya, S. P., On m-paracompact spaces, Math. Ann. 181 (1969), 119133.Google Scholar
27. Sorgenfrey, R. H., On the topological product of paracompact spaces, Bull. Amer. Math. Soc. 53 (1947), 631632.Google Scholar
28. Stone, A. H., Paracompactness and product spaces, Bull. Amer. Math. Soc. 54 (1948), 977982.Google Scholar
29. Worrell, J. M., Jr. and Wicke, H. H., Characterizations of developable topological spaces, Can. J. Math. 17 (1965), 820830.Google Scholar
30. Zenor, P., On countable paracompactness and normality, Prace Mat. 13 (1969), 2332.Google Scholar