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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
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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

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