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Old and new results on quasi-uniform extension
Császár, Ákos
1999
Abstract
According to $\left[17\right]$ or $\left[12\right]$, $\mathcal{U}$
is a quasi-uniformity on a set X if it's a filter on $X\times X$,
the diagonal $\Delta=\left\{ \left(x,x\right):x\epsilon X\right\} \subset U$
for U $\epsilon\; U$ (i.e. $\mathcal{U}$ is composed of entourages
on X), and, for each U $\epsilon\;\mathcal{U}$, there is U' $\epsilon\;\mathcal{U}$
such that U'$^{2}$=U' o U'=$\left\{ \left(x,z\right):\exists y\;\textrm{with}\;\left(x,y\right),\left(y,z\right)\epsilon U'\right\} \subset U.$
The restriction $\mathcal{U}\mid X_{0}$ to $X_{0}\subset X$ of the
quasi-uniformity $\mathcal{U}$ on X is composed of the sets $\mathcal{U}\mid X_{0}=U\cap\left(X_{0}\times X_{0}\right)$
for U $\epsilon\; U$; it is a quasi-uniformity on X$_{0}$. Let Y
$\supset$X, $\mathcal{U}$ be a quasi-uniformity on Y; $\mathcal{W}$
is an extension of the quasi-uniformity $\mathcal{U}$ on X if $\mathcal{W}\mid X\mathcal{=U}$.
The purpose of the present paper is to give a survey on results, due
mainly to Hungarian topologists, concerning extensions of quasi-uniformities.
Series
Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
30 (1999) suppl.
Publisher
Università degli Studi di Trieste. Dipartimento di Scienze Matematiche
Source
Á. Császár, "Old and new results on quasi-uniform extension", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 30 (1999) suppl., pp. 75-85.
Languages
en
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