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Keywords:
Hausdorff-Bourbaki quasi-uniformity; hyperspace; locally compact; cofinally complete; uniformly locally compact; co-uniformly locally compact
Hausdorff-Bourbaki quasi-uniformity; hyperspace; locally compact; cofinally complete; uniformly locally compact; co-uniformly locally compact
Summary:
We characterize those Tychonoff quasi-uniform spaces $(X,\mathcal {U})$ for which the Hausdorff-Bourbaki quasi-uniformity is uniformly locally compact on the family $\mathcal {K}_{0}(X)$ 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 $\mathcal {K}_{0}(X)$ 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 $\sigma $-compact if and only if its (lower) semicontinuous quasi-uniformity is co-uniformly locally compact. A characterization of those Hausdorff quasi-uniform spaces $(X,\mathcal {U})$ for which the Hausdorff-Bourbaki quasi-uniformity is co-uniformly locally compact on $\mathcal {K}_{0}(X)$ is obtained.
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