The Lindelöf property and pseudo-${\aleph }_1$-compactness in spaces and topological groups

@article{Hernández2008,
abstract = {We introduce and study, following Z. Frol’ık, the class $\mathcal \{B\}(\mathcal \{P\})$ of regular $P$-spaces $X$ such that the product $X\times Y$ is pseudo-$\aleph _1$-compact, for every regular pseudo-$\aleph _1$-compact $P$-space $Y$. We show that every pseudo-$\aleph _1$-compact space which is locally $\mathcal \{B\}(\mathcal \{P\})$ is in $\mathcal \{B\}(\mathcal \{P\})$ and that every regular Lindelöf $P$-space belongs to $\mathcal \{B\}(\mathcal \{P\})$. It is also proved that all pseudo-$\aleph _1$-compact $P$-groups are in $\mathcal \{B\}(\mathcal \{P\})$. The problem of characterization of subgroups of $\mathbb \{R\}$-factorizable (equivalently, pseudo-$\aleph _1$-compact) $P$-groups is considered as well. We give some necessary conditions on a topological $P$-group to be a subgroup of an $\mathbb \{R\}$-factorizable $P$-group and deduce that there exists an $\omega $-narrow $P$-group that cannot be embedded as a subgroup into any $\mathbb \{R\}$-factorizable $P$-group. The class of $\sigma $-products of second-countable topological groups is especially interesting. We prove that all subgroups of the groups in this class are perfectly $\kappa $-normal, $\mathbb \{R\}$-factorizable, and have countable cellularity. If, in addition, $H$ is a closed subgroup of a $\sigma $-product of second-countable groups, then $H$ is an Efimov space and satisfies $\operatorname\{cel\}_\omega (H)\le \omega $.},
author = {Hernández, Constancio, Tkachenko, Mihail G.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {pseudo-$\aleph _1$-compact space; $\mathbb \{R\}$-factorizable group; cellularity; $\sigma $-product; -factorizable group; products},
language = {eng},
number = {4},
pages = {677-692},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The Lindelöf property and pseudo-$\aleph _1$-compactness in spaces and topological groups},
url = {http://eudml.org/doc/250494},
volume = {49},
year = {2008},
}

TY - JOUR
AU - Hernández, Constancio
AU - Tkachenko, Mihail G.
TI - The Lindelöf property and pseudo-$\aleph _1$-compactness in spaces and topological groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2008
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 49
IS - 4
SP - 677
EP - 692
AB - We introduce and study, following Z. Frol’ık, the class $\mathcal {B}(\mathcal {P})$ of regular $P$-spaces $X$ such that the product $X\times Y$ is pseudo-$\aleph _1$-compact, for every regular pseudo-$\aleph _1$-compact $P$-space $Y$. We show that every pseudo-$\aleph _1$-compact space which is locally $\mathcal {B}(\mathcal {P})$ is in $\mathcal {B}(\mathcal {P})$ and that every regular Lindelöf $P$-space belongs to $\mathcal {B}(\mathcal {P})$. It is also proved that all pseudo-$\aleph _1$-compact $P$-groups are in $\mathcal {B}(\mathcal {P})$. The problem of characterization of subgroups of $\mathbb {R}$-factorizable (equivalently, pseudo-$\aleph _1$-compact) $P$-groups is considered as well. We give some necessary conditions on a topological $P$-group to be a subgroup of an $\mathbb {R}$-factorizable $P$-group and deduce that there exists an $\omega $-narrow $P$-group that cannot be embedded as a subgroup into any $\mathbb {R}$-factorizable $P$-group. The class of $\sigma $-products of second-countable topological groups is especially interesting. We prove that all subgroups of the groups in this class are perfectly $\kappa $-normal, $\mathbb {R}$-factorizable, and have countable cellularity. If, in addition, $H$ is a closed subgroup of a $\sigma $-product of second-countable groups, then $H$ is an Efimov space and satisfies $\operatorname{cel}_\omega (H)\le \omega $.
LA - eng
KW - pseudo-$\aleph _1$-compact space; $\mathbb {R}$-factorizable group; cellularity; $\sigma $-product; -factorizable group; products
UR - http://eudml.org/doc/250494
ER -