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 ${\tilde{\mathcal{U}}}_H$ of their bicompletion $(\tilde{X},\tilde{\mathcal{U}})$ on ${\mathcal{P}}_0(\tilde{X})$ is bicomplete.