Affiliation:
1. 1 Universidad Politécnica de Valencia Escuela de Caminos, Departmento de Matemática Aplicada, IMPA-UPV 46071 Valencia Spain
2. 2 Universidad de Almería Área de Geometría y Topología, Facultad de Ciencias Experimentales 04120 Almería Spain
Abstract
By a *-compactification of a
T0
quasi-uniform space (
X, U
) we mean a compact
T0
quasi-uniform space (
Y, V
) that has a
T
(
V
∨
V−1
)-dense subspace quasi-isomorphic to (
X, U
). We prove that (
X, U
) has a *-compactification if and only if its
T0
biocompletion \documentclass{aastex}
\usepackage{amsbsy}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{mathrsfs}
\usepackage{pifont}
\usepackage{stmaryrd}
\usepackage{textcomp}
\usepackage{upgreek}
\usepackage{portland,xspace}
\usepackage{amsmath,amsxtra}
\usepackage{bbm}
\pagestyle{empty}
\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
$$({\tilde X},\tilde {\mathcal{U}})$$
\end{document} is compact. We also show that, in this case, \documentclass{aastex}
\usepackage{amsbsy}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{mathrsfs}
\usepackage{pifont}
\usepackage{stmaryrd}
\usepackage{textcomp}
\usepackage{upgreek}
\usepackage{portland,xspace}
\usepackage{amsmath,amsxtra}
\usepackage{bbm}
\pagestyle{empty}
\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
$$({\tilde X},\tilde {\mathcal{U}})$$
\end{document} is the maximal *-compactification of (
X, U
) and (
X
∪
G
(
X
), \documentclass{aastex}
\usepackage{amsbsy}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{mathrsfs}
\usepackage{pifont}
\usepackage{stmaryrd}
\usepackage{textcomp}
\usepackage{upgreek}
\usepackage{portland,xspace}
\usepackage{amsmath,amsxtra}
\usepackage{bbm}
\pagestyle{empty}
\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
$$\tilde {\mathcal{U}}$$
\end{document}|
X
∪
G
(
X
)
) is its minimal *-compactification, where
G
(
X
) is the set of all points of \documentclass{aastex}
\usepackage{amsbsy}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{mathrsfs}
\usepackage{pifont}
\usepackage{stmaryrd}
\usepackage{textcomp}
\usepackage{upgreek}
\usepackage{portland,xspace}
\usepackage{amsmath,amsxtra}
\usepackage{bbm}
\pagestyle{empty}
\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
$$\tilde X$$
\end{document} which are
T
(\documentclass{aastex}
\usepackage{amsbsy}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{mathrsfs}
\usepackage{pifont}
\usepackage{stmaryrd}
\usepackage{textcomp}
\usepackage{upgreek}
\usepackage{portland,xspace}
\usepackage{amsmath,amsxtra}
\usepackage{bbm}
\pagestyle{empty}
\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
$$\tilde {\mathcal{U}}$$
\end{document})-closed (we remark that as partial order of *-compactifications we use the inverse of the partial order used for
T2
compactifications of Tychonoff spaces). Applications of our results to some examples in theoretical computer science are given.
Reference23 articles.
1. PCF extended with real numbers. Real numbers and computers;Escardo M.;Theoret. Comput. Sci.,1996
2. Completeness using pairs of filters;Fletcher P.;Topology Appl.,1992
3. Compactifications of totally bounded quasi-uniform spaces;Fletcher P.;Glasgow Math. J.,1986
4. Gierz, G., Hofmann, H., Keimel, K., Lawson, J. D., Mislove, M. W. and Scott, D. S. , Continuous Lattices and Domains , Encyclopedia of Mathematics and its Applications 93, Cambridge Univ. Press (2003). MR 2004h :06001
Cited by
2 articles.
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