Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-06-09T02:08:05.776Z Has data issue: false hasContentIssue false
  • English
  • Français

Even Covers and Collectionwise Normal Spaces

Published online by Cambridge University Press:  20 November 2018

H. L. Shapiro  and
F. A. Smith
H. L. Shapiro
Affiliation:
Northern Illinois University, De Kalb, Illinois
F. A. Smith
Affiliation:
Kent State University, Kent, 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.

The concept of an even cover is introduced early in elementary topology courses and is known to be valuable. Among other facts it is known that X is paracompact if and only if every open cover of X is even. In this paper we introduce the concept of an n-even cover and show its usefulness. Using n-even we define an embedding that on closed subsets is equivalent to collectionwise normal. We also give sufficient conditions for a point finite open cover to have a locally finite refinement and also sufficient conditions for this refinement to be even. Finally we show that the collection of all neighborhoods of the diagonal of X is a uniformity if and only if every even cover is normal. This last result is particularly interesting in light of the fact that every normal open cover is even.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 3 , 01 June 1978 , pp. 466 - 473
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Alo, R. A. and Shapiro, H. L., Normal topological spaces (Cambridge University Press, Cambridge, Great Britain, 1974).Google Scholar
2. Bing, R. H., Metrization of topological spaces, Can. J. Math. 3 (1951), 175–186.Google Scholar
3. Cohen, H. J., Sur un problème de M. Dieudonné, C. R. Acad. Sci. Paris 234 (1952), 290–292.Google Scholar
4. Egorov, V. I., On the metric dimension of point sets, Mat. Sb. 48 (1959), 227–250. (Russian)Google Scholar
5. Hayashi, Y., On countably metacompact spaces, Bull. Univ. Prefecture Ser. A, 8 (1959/60), 161–164.Google Scholar
6. Isbell, J. R., Uniform spaces (Amer. Math. Soc, Providence, 1964).Google Scholar
7. Michael, E., Point-finite and locally finite coverings, Can. J. Math. 7 (1955), 275–279.Google Scholar
8. Smith, J. C., Jr., Characterizations of metric-dependent dimension functions, Proc. Amer. Math. Soc. 19 (1968), 1264–1269.Google Scholar