Idempotency of Extensions via the Bicompletion |
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Authors: | G. C. L. Brümmer H. -P. A. Künzi |
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Affiliation: | (1) Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701, South Africa |
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Abstract: | Let Top 0 be the category of topological T 0-spaces, QU 0 the category of quasi-uniform T 0-spaces, T : QU 0 → Top 0 the usual forgetful functor and K : QU 0 → QU 0 the bicompletion reflector with unit k : 1 → K. Any T-section F : Top 0 → QU 0 is called K-true if KF = FTKF, and upper (lower) K-true if KF is finer (coarser) than FTKF. The literature considers important T-sections F that enjoy all three, or just one, or none of these properties. It is known that T(K,k)F is well-pointed if and only if F is upper K-true. We prove the surprising fact that T(K,k)F is the reflection to Fix(TkF) whenever it is idempotent. We also prove a new characterization of upper K-trueness. We construct examples to set apart some natural cases. In particular we present an upper K-true F for which T(K,k)F is not idempotent, and a K-true F for which the coarsest associated T-preserving coreflector in QU 0 is not stable under K. We dedicate this paper to the memory of Sérgio de Ornelas Salbany (1941–2005). |
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Keywords: | lower K-true upper K-true K-true lower K-stable idempotent functorial quasi-uniformity superrigid space |
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