On Uniformly Locally Compact Quasi-Uniform Hyperspaces

We characterize those Tychonoff quasi-uniform spaces $$\left( {X,U} \right)$$ for which the Hausdorff-Bourbaki quasi-uniformity is uniformly locally compact on the family $$K_{\text{0}} \left( X \right)$$ of nonempty compact subsets of X. We deduce, among other results, that the Hausdorff-Bourbaki quasi-uniformity of the locally finite quasi-uniformity of a Tychonoff space Xis uniformly locally compact on $$K_{\text{0}} \left( X \right)$$ if and only if Xis paracompact and locally compact. We also introduce the notion of a co-uniformly locally compact quasi-uniform space and show that a Hausdorff topological space is σ-compact if and only if its (lower) semi-continuous quasi-uniformity is co-uniformly locally compact. A characterization of those Hausdorff quasi-uniform spaces $$\left( {X,U} \right)$$ for which the Hausdorff-Bourbaki quasi-uniformity is co-uniformly locally compact on $$K_{\text{0}} \left( X \right)$$ is obtained.