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Rich Proximities and Compactifications

Published online by Cambridge University Press:  20 November 2018

Stephan C. Carlson
Stephan C. Carlson*
Affiliation:
University of North Dakota, Grand Forks, North Dakota
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Each Hausdorff compactification of a given Tychonoff space is the Smirnov compactification associated with a compatible proximity on the space. Also each realcompactification of a given Tychonoff space is the underlying topological space of the completion of a compatible uniformity on the space. But if T is a realcompactification of a Tychonoff space X which is contained in a particular compactification Z of X, then it is not always possible to find a compatible uniformity on X such that T is the underlying topological space of the completion of (X, ) and induces the proximity on X associated with Z. We shall call a Hausdorff compactification Z of a Tychonoff space X a rich compactification of X (and the associated proximity on X a rich proximity) if every realcompactification of X contained in Z can be obtained as the underlying topological space of the completion of a compatible uniformity on X which induces the proximity on X associated with Z.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 2 , 01 April 1982 , pp. 319 - 348
Copyright
Copyright © Canadian Mathematical Society 1981

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