Sobrification and bicompletion of totally bounded quasi-uniform spaces
- 1 March 1987
- journal article
- research article
- Published by Cambridge University Press (CUP) in Mathematical Proceedings of the Cambridge Philosophical Society
- Vol. 101 (2) , 237-247
- https://doi.org/10.1017/s0305004100066597
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 .Keywords
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