Charles University in Prague, Faculty of Mathematics and Physics
Abstract
summary: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 ℜL 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 U on ℜL such that (ℜL,U) is a compactification of (L,U). Moreover, the Cauchy spectrum of the uniform frame (Fr(U∗),U∗) can be viewed as the spectrum of the bicompletion of (L,U)