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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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