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2. On $\sigma$-discrete Borel mappings via quasi-metrics
- Creator:
- Künzi, Hans-Peter A. and Wajch, Eliza
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- quasi-metric, continuous map, Borel map, $\sigma $-discrete map, $\sigma $-discretely decomposable family, absolutely Borel set, and absolutely analytic space
- Language:
- English
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
3. On uniformly locally compact quasi-uniform hyperspaces
- Creator:
- Künzi, Hans-Peter A., Romaguera, Salvador , and Sánchez-Granero, M. A.
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Hausdorff-Bourbaki quasi-uniformity, hyperspace, locally compact, cofinally complete, uniformly locally compact, and co-uniformly locally compact
- Language:
- English
- Description:
- 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.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public