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Totally bounded frame quasi-uniformities. (English) Zbl 0786.54028
Summary: This paper considers totally bounded quasi-uniformities and quasi- proximities for frames and shows that for a given quasi-proximity \(\triangleleft\) on a frame \(L\) there is a totally bounded quasi- uniformity on \(L\) that is the coarsest quasi-uniformity, and the only totally bounded quasi-uniformity, that determines \(\triangleleft\). The construction due to B. Banaschewski and A. Pultr [Math. Proc. Camb. Philos. Soc. 108, No. 1, 63-78 (1990; Zbl 0733.54020)] of the Cauchy spectrum \(\psi L\) and the compactification \({\mathfrak R} L\) of a uniform frame \((L,U)\) are meaningful for quasi-uniform frames. If \(U\) is a totally bounded quasi-uniformity on a frame \(L\), there is a totally bounded quasi-uniformity \(\overline U\) on \({\mathfrak R}L\) such that \(({\mathfrak R} L,\overline U)\) is a compactification of \((L,U)\). Moreover, the Cauchy spectrum of the uniform frame \((\text{Fr}(U^*),U^*)\) can be viewed as the spectrum of the bicompletion of \((L,U)\).

MSC:
54E15 Uniform structures and generalizations
06D20 Heyting algebras (lattice-theoretic aspects)
54E05 Proximity structures and generalizations
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
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