Totally bounded frame quasi-uniformities

@article{Fletcher1993,
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 $(L, \{\mathbf \{U\}\})$ are meaningful for quasi-uniform frames. If $\{\mathbf \{U\}\}$ is a totally bounded quasi-uniformity on a frame $L$, there is a totally bounded quasi-uniformity $\overline\{\{\mathbf \{U\}\}\}$ on $\Re L$ such that $(\Re L, \overline\{\{\mathbf \{U\}\}\})$ is a compactification of $(L,\{\mathbf \{U\}\})$. Moreover, the Cauchy spectrum of the uniform frame $(Fr(\{\mathbf \{U\}\}^\{\ast \}), \{\mathbf \{U\}\}^\{\ast \})$ can be viewed as the spectrum of the bicompletion of $(L,\{\mathbf \{U\}\})$.},
author = {Fletcher, Peter, Hunsaker, Worthen N., Lindgren, William F.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {frame; uniform frame; quasi-uniform frame; quasi-proximity; totally bounded quasi-uniformity; uniformly regular ideal; compactification; bicompletion; quasi-proximity; uniformly regular ideal; uniform frame; quasi-uniform frames; totally bounded quasi-uniformity; compactification; bicompletion},
language = {eng},
number = {3},
pages = {529-537},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Totally bounded frame quasi-uniformities},
url = {http://eudml.org/doc/247453},
volume = {34},
year = {1993},
}

TY - JOUR
AU - Fletcher, Peter
AU - Hunsaker, Worthen N.
AU - Lindgren, William F.
TI - Totally bounded frame quasi-uniformities
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 3
SP - 529
EP - 537
AB - 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 $(L, {\mathbf {U}})$ are meaningful for quasi-uniform frames. If ${\mathbf {U}}$ is a totally bounded quasi-uniformity on a frame $L$, there is a totally bounded quasi-uniformity $\overline{{\mathbf {U}}}$ on $\Re L$ such that $(\Re L, \overline{{\mathbf {U}}})$ is a compactification of $(L,{\mathbf {U}})$. Moreover, the Cauchy spectrum of the uniform frame $(Fr({\mathbf {U}}^{\ast }), {\mathbf {U}}^{\ast })$ can be viewed as the spectrum of the bicompletion of $(L,{\mathbf {U}})$.
LA - eng
KW - frame; uniform frame; quasi-uniform frame; quasi-proximity; totally bounded quasi-uniformity; uniformly regular ideal; compactification; bicompletion; quasi-proximity; uniformly regular ideal; uniform frame; quasi-uniform frames; totally bounded quasi-uniformity; compactification; bicompletion
UR - http://eudml.org/doc/247453
ER -