Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T00:52:27.001Z Has data issue: false hasContentIssue false
  • English
  • Français

Spaces with a Unique Uniformity

Published online by Cambridge University Press:  20 November 2018

Richard H. Warren
Richard H. Warren*
Affiliation:
Military Programs Department, General Electric Company, King of Prussia, Pennsylvania
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 major results in this paper are nine characterizations of completely regular spaces with a unique compatible uniformity. All prior results of this type assumed that the space is Tychonoff (i.e., completely regular and Hausdorff) until the appearance of a companion paper [9] which began this study. The more important characterizations use quasi-uniqueness of R1-compactifications which relate to uniqueness of T2-comPactifications. The features of the other characterizations are: (i) compact subsets linked to Cauchy filters, (ii) C- and C*-embeddings, and (iii) lifting continuous maps to uniformly continuous maps.

Section 2 contains information on T0-identification spaces which we will use later in the paper. In Section 3 several properties of uniform identification spaces are developed so that they can be used later. The nine characterizations are established in Section 4. Also it is shown that a space with a unique compatible uniformity is normal if and only if each of its closed subspaces has a unique compatible uniformity.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 6 , 01 December 1983 , pp. 1001 - 1009
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Davis, A. S., Indexed systems of neighborhoods for general topological spaces, Amer. Math. Monthly 68 (1961), 886–893.Google Scholar
2. Dickinson, A., Compactness conditions and uniform structures, Amer. J. Math. 75 (1953), 224–228.Google Scholar
3. Dickman, R. F. Jr., Porter, J. R. and Rubin, L. R., Completely regular absolutes and projective objects, Pacific J. Math. 94 (1981), 277–295.Google Scholar
4. Doss, R., On uniform spaces with a unique structure, Amer. J. Math. 71 (1949), 19–23.Google Scholar
5. Gillman, L. and Jerison, M., Rings of continuous functions (Van Nostrand, Princeton, 1960).CrossRefGoogle Scholar
6. Heldermann, N. C., Develop ability and some new regularity axioms, Can. J. Math. 33 (1981), 641–663.Google Scholar
7. Kent, D. C. and Richardson, G. D., Completely regular and ω-regular spaces, Proc. AMS 82 (1981), 649–652.Google Scholar
8. Newns, W. F., Uniform spaces with unique structure, Amer. J. Math. 79 (1957), 48–52.Google Scholar
9. Warren, R. H., Identification spaces and unique uniformity, Pacific J. Math. 95 (1981), 483–492.Google Scholar