Totally bounded frame quasi-uniformities

Fletcher, P. ; Hunsaker, W. ; Lindgren, W.

Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 3, pp. 529-537.

Résumé

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 $(L,{\bold U})$.

Détail

Classification : 06D20, 18B35, 54D35, 54E05, 54E15
Keywords: frame; uniform frame; quasi-uniform frame; quasi-proximity; totally bounded quasi-uniformity; uniformly regular ideal; compactification; bicompletion