The bicompletion of the Hausdorff quasi-uniformity

We study conditions under which the Hausdorff quasi-uniformity U H of a quasi-uniform space ( X , U ) on the set P 0 ( X ) of the nonempty subsets of X is bicomplete. 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 , U ) for which the Hausdorff quasi-uniformity U ˜ H of their bicompletion ( X ˜ , U ˜ ) on P 0 ( X ˜ ) is bicomplete.