Old and new results on quasi-uniform extension

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.