Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-14T17:54:55.829Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on Extending Locally Finite Collections

Published online by Cambridge University Press:  20 November 2018

H. L. Shapiro  and
F. A. Smith
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.

Recently there has been a great deal of interest in extending refinements of locally finite and point finite collections on subsets of certain topological spaces. In particular the first named author showed that a subset S of a topological space X is P-embedded in X if and only if every locally finite cozero-set cover on S has a refinement that can be extended to a locally finite cozero-set cover of X. Since then many authors have studied similar types of embeddings (see [1], [2], [3], [4], [6], [8], [9], [10], [11], and [12]). Since the above characterization of P-embedding is equivalent to extending continuous pseudometrics from the subspace S up to the whole space X, it is natural to wonder when can a locally finite or a point finite open or cozero-set cover on S be extended to a locally finite or point-finite open or cozero-set cover on X.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 1 , 01 March 1976 , pp. 117 - 119
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Alo, R. A. and Shapiro, H. L., Countably paracompact, normal, and collectionwise normal spaces, Indag. Math. 35 (1973), 347–351.Google Scholar
2. Alo, R. A., Normal Topological Spaces (London, Cambridge University Press, 1974).Google Scholar
3. Gantner, T. E., Extensions of uniformly continuous pseudometrics, Trans. Amer. Math. Soc. 132 (1968), 147–157.Google Scholar
4. Gantner, T. E., Extensions of uniformities, Fund. Math. 66 (1970), 263–281.Google Scholar
5. Katetov, M., Extension of locally finite covers, Colloq. Math. 6 (1958), 145-151 (Russian).Google Scholar
6. Krajewski, L. L., On expanding locally finite collections, Canad. J. Math. 23 (1971), 58–68.Google Scholar
7. Michael, E., Point-finite and locally finite covers, Canad. J. Math. 7 (1955), 275–279.Google Scholar
8. Sennott, L. I., Extending point-finite covers, Proc. Third Prague Topological Symposium (1971), 393–397.Google Scholar
9. Shapiro, H. L., Extensions of pseudometrics, Canad. J. Math. 18 (1966), 981–998.Google Scholar
10. Shapiro, H. L., More on extending continuous pseudometrics, Canad. J. Math., 22 (1970), 984–993.Google Scholar
11. Smith, J. C. and Krajewski, L. L., Expandability and collectionwise normality, Trans. Amer. Math. Soc. 160(1971), 437–51.Google Scholar
12. Smith, J. C. and Nichols, J. C., Embedding characterizations for expandable spaces, Duke Math. J. 39(1972), 489–496.Google Scholar