On uniformly locally compact quasi-uniform hyperspaces

Künzi, Hans-Peter A., Romaguera, Salvador, and Sánchez-Granero, M. A.. "On uniformly locally compact quasi-uniform hyperspaces." Czechoslovak Mathematical Journal 54.1 (2004): 215-228. <http://eudml.org/doc/30851>.

@article{Künzi2004,
abstract = {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.},
author = {Künzi, Hans-Peter A., Romaguera, Salvador, Sánchez-Granero, M. A.},
journal = {Czechoslovak Mathematical Journal},
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},
language = {eng},
number = {1},
pages = {215-228},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On uniformly locally compact quasi-uniform hyperspaces},
url = {http://eudml.org/doc/30851},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Künzi, Hans-Peter A.
AU - Romaguera, Salvador
AU - Sánchez-Granero, M. A.
TI - On uniformly locally compact quasi-uniform hyperspaces
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 1
SP - 215
EP - 228
AB - 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.
LA - eng
KW - 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
UR - http://eudml.org/doc/30851
ER -