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 $\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 constructions due to B. Banaschewski and A. Pultr of the Cauchy spectrum $\psi L$ and the compactification $\Re L$ of a uniform frame $(L, {\bold U})$ are meaningful for quasi-uniform frames. If ${\bold U}$ is a totally bounded quasi-uniformity on a frame $L$, there is a totally bounded quasi-uniformity $\overline {{\bold U}}$ on $\Re L$ such that $(\Re L, \overline {{\bold U}})$ is a compactification of $(L,{\bold U})$. Moreover, the Cauchy spectrum of the uniform frame $(Fr({\bold U}^{*}), {\bold U}^{*})$ can be viewed as the spectrum of the bicompletion of $(L,{\bold U})$.

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