Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-16T01:24:51.745Z Has data issue: false hasContentIssue false
  • English
  • Français

On the “Essential Metrization” of Uniform Spaces

Published online by Cambridge University Press:  20 November 2018

Elias Zakon
Elias Zakon*
Affiliation:
University of Windsor, Canada, Summer Research Institute of the Canadian Mathematical Congress
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 notion of “essential metrization” was introduced in (8) where it was used to obtain some extensions of the theorems of Egoroff and Lusin on measurable functions. In the present note we shall further develop the theory of essential metrization, in its own right.

As is well known, sets of measure zero (“null sets“) may be disregarded in many problems of measure theory. Hence the usual topological prerequisites of such problems are actually too restrictive and may be replaced by what could be called “topology modulo null-sets,” imitating such notions as “approximate continuity,” “essential supremum,” etc. Thus we consider spaces in which certain topological properties (such as metrizability, separability, etc.) hold not in the usual sense but only “essentially,” i.e. to within some null sets, as defined below.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 1224 - 1236
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Bourbaki, N., Topologie générale, chaps. 1, II, IX (Paris, 1960).Google Scholar
2. Colmez, J., Sur divers problèmes concernant les espaces topologiques, Portugal Math. 6 (1947), 119–224.Google Scholar
3. Doss, R., Sur la théorie de l'écart abstrait de M. Fréchet, Bull. Sci. Math. (2), 71 (1947), 110–122.Google Scholar
4. Isbell, J. R., Uniform spaces (Providence, 1964).Google Scholar
5. Kelley, J., General topology (New York, 1955).Google Scholar
6. Munroe, M. E., Introduction to measure and integration (Reading, Mass., 1959).Google Scholar
7. Weil, A., Sur les espaces à structure uniforme et sur la topologie générale (Paris, 1938).Google Scholar
8. Zakon, E., On “essentially metrizable” spaces and on measurable functions with values in such spaces, Trans. Amer. Math. Soc., 119 (1965), 443–453.Google Scholar
9. On uniform spaces with quasi-nested base (to appear).Google Scholar