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P. Fletcher, W. Hunsaker, W. Lindgren Totally bounded frame quasi-uniformities

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P. Fletcher, W. Hunsaker, W. Lindgren Totally bounded frame quasi-uniformities

Comment.Math.Univ.Carolinae 34,3 (1993) 529-537.

Abstract: This paper considers totally bounded quasi-uniformities and quasi- proximities for frames and shows that for a given quasi-proximity / 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/. The constructions due to B. Banaschewski and A. Pultr of the Cauchy spectrum ψL and the compactification <Lof a uniform frame (L,U) are meaningful for quasi- uniform frames. IfUis a totally bounded quasi-uniformity on a frameL, there is a totally bounded quasi-uniformityUon<Lsuch that (<L,U) is a compactification of (L,U). Moreover, the Cauchy spectrum of the uniform frame (F r(U∗),U∗) can be viewed as the spectrum of the bicompletion of (L,U).

Keywords: frame, uniform frame, quasi-uniform frame, quasi-proximity, totally bounded quasi-uniformity, uniformly regular ideal, compactification, bicompletion AMS Subject Classification: 6D20, 18B35, 54D35, 54E05, 54E15

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