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

Abstract

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 $\left[2\right]$