Geodesic
Totally bounded frame quasi-uniformities
Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 3, pp. 529-537 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $(L,{\bold U})$.
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 $(L,{\bold U})$.
Classification : 06D20, 18B35, 54D35, 54E05, 54E15
Keywords: frame; uniform frame; quasi-uniform frame; quasi-proximity; totally bounded quasi-uniformity; uniformly regular ideal; compactification; bicompletion 
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     author = {Fletcher, P. and Hunsaker, W. and Lindgren, W.},
     title = {Totally bounded frame quasi-uniformities},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {529--537},
     year = {1993},
     volume = {34},
     number = {3},
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     zbl = {0786.54028},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1993_34_3_a14/}
}
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Fletcher, P.; Hunsaker, W.; Lindgren, W. Totally bounded frame quasi-uniformities. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 3, pp. 529-537. http://geodesic.mathdoc.fr/item/CMUC_1993_34_3_a14/