Old and new results on quasi-uniform extension

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

On J. Deák's construction for quasi-uniform extensions

Let (X,$\mathcal{U}$) be a quasi-umform space, Y $\supset$ X, $\mathcal{T}$ a topology on Y. An extension compatible with ($\mathcal{U}$,$\mathcal{T}$) is a quasiuniformity $\mathcal{W}$ on Y such that the restriction $\mathcal{W}\mid$ X of $\mathcal{W}$ to X coincides with $\mathcal{U}$ and the topology $\mathcal{W}^{tp}$ induced by $\mathcal{W}$ equals $\mathcal{T}$. The paper $\left[1\right]$ contains a construction of such extensions. The purpose of the present paper is to give some applications of the result in $\left[1\right]$. Without explicit mention of the contrary, we shall use the terminology and notation of $\left[2\right]$

Merotopies associated with quasi-uniformities

[EN] To an arbitrary quasi-uniformity on the set X, a merotopy on X is assigned. There are results concerning the question whether this merotopy is compatible with the topology induced by the quasi-uniformity end whether the closure operation induced by the merotopy, admits a compatible uniformity. More precise results are obtained in the case of transitive quasi-uniformities.Research supported by Hungarian Foundation for Scienti c Research, grant No. T0320Császár, Á. (2000). Merotopies associated with quasi-uniformities. Applied General Topology. 1(1):1-12. https://doi.org/10.4995/agt.2000.3020SWORD1121

Cauchy structures and contiguities

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