On uniformly locally compact quasi-uniform hyperspaces
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We characterize those Tychonoff quasi-uniform spaces (X, U) for which the Hausdorff-Bourbaki quasi-uniformity is uniformly locally compact on the family K0(X) 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 X is uniformly locally compact on K0(X) if and only if X is 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) semicontinuous quasi-uniformity is co-uniformly locally compact. A characterization of those Hausdorff quasi-uniform spaces (X, U) for which the Hausdorff-Bourbaki quasi-uniformity is co-uniformly locally compact on K0(X) is obtained.