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}
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$$({\tilde X},\tilde {\mathcal{U}})$$
\end{document} is compact. We also show that, in this case, \documentclass{aastex}
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$$({\tilde X},\tilde {\mathcal{U}})$$
\end{document} is the maximal *-compactification of (
X, U
) and (
X
∪
G
(
X
), \documentclass{aastex}
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\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}
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$$\tilde X$$
\end{document} which are
T
(\documentclass{aastex}
\usepackage{amsbsy}
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\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.
On compactification of mappings