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Sobrification and bicompletion of totally bounded quasi-uniform spaces

Published online by Cambridge University Press:  24 October 2008

H. P. A. Künzi  and
G. C. L. Brümmer
H. P. A. Künzi
Affiliation:
Department of Mathematics, University of Berne, Berne, Switzerland
G. C. L. Brümmer
Affiliation:
Department of Mathematics, University of Cape Town, Rondebosch, South Africa
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Abstract

We observe that if is a compatible totally bounded quasi-uniformity on a T0-space (X,), then the bicompletion of (X, ) is a strongly sober, locally quasicompact space. It follows that the b-closure S of (X, ) in is homeomorphic to the sobrification of the space (X, ). We prove that S is equal to if and only if (X, ) is a core-compact space in which every ultrafilter has an irreducible convergence set and is the coarsest quasi-uniformity compatible with . If is the Pervin quasi-uniformity on X, then S is equal to if and only if X is hereditarily quasicompact, or equivalently, is the Pervin quasi-uniformity on .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 2 , March 1987 , pp. 237 - 247
Copyright
Copyright © Cambridge Philosophical Society 1987

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