The bicompletion of the Hausdorff quasi-uniformity

Abstract

We study conditions under which the Hausdorff quasi-uniformity ${\mathcal U}_H$ of a quasi-uniform space $(X,{\mathcal U})$ on the set ${\mathcal P}_0(X)$ of the nonempty subsets of $X$ is bicomplete. Indeed we present an explicit method to construct the bicompletion of the $T_0$-quotient of the Hausdorff quasi-uniformity of a quasi-uniform space. It is used to find a characterization of those quasi-uniform $T_0$-spaces $(X,{\mathcal U})$ for which the Hausdorff quasi-uniformity $\widetilde{{\mathcal U}}_H$ of their bicompletion $(\widetilde{X},{\widetilde{\mathcal U}})$ on ${\mathcal P}_0(\widetilde{X})$ is bicomplete.