Totally bounded frame quasi-uniformities

Abstract

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 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 $(Fr({\bold U}^{\ast }), {\bold U}^{\ast })$ can be viewed as the spectrum of the bicompletion of $(L,{\bold U})$