The well-known Isbell-Burdick Theorem asserts that for a uniform space \((X,\mathcal{U})\) the Hausdorff uniformity on the set \(\mathcal{P}_0(X)\) of the nonempty subsets of \(X\) is complete if and only if each stable filter on \((X,\mathcal{U})\) has a cluster point. In the setting of quasi-uniform spaces the corresponding problem under which conditions the Hausdorff quasi-uniformity on \(\mathcal{P}_0(X)\) is bicomplete has not yet such a satisfactory answer. In [Topol. Proc. 20, 161–183 (1995; Zbl 0876.54022)], H.-P. Künzi and C. Ryser provided some conditions. In particular, these authors observed that the Hausdorff quasi-uniformity of a totally bounded and bicomplete quasi-uniformity is (totally bounded and) bicomplete.
The present paper continues the search for conditions under which the Hausdorff quasi-uniformity \(\mathcal{U}_H\) of a quasi-uniform space \((X,\mathcal{U})\) on \(\mathcal{P}_0(X)\) is bicomplete. The authors present a general method to construct the bicompletion of the \(T_0\)-quotient of the Hausdorff quasi-uniformity of a quasi-uniform space, and use it 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.