| Title:
|
The Lindelöf property and pseudo-$\aleph_1$-compactness in spaces and topological groups (English) |
| Author:
|
Hernández, Constancio |
| Author:
|
Tkachenko, Mikhail |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
49 |
| Issue:
|
4 |
| Year:
|
2008 |
| Pages:
|
677-692 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We introduce and study, following Z. Frol'{\i}k, the class $\Cal B(\Cal P)$ 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 $\Cal B(\Cal P)$ is in $\Cal B(\Cal P)$ and that every regular Lindelöf $P$-space belongs to $\Cal B(\Cal P)$. It is also proved that all pseudo-$\aleph_1$-compact $P$-groups are in $\Cal B(\Cal P)$. The problem of characterization of subgroups of $\Bbb R$-factor\-izable (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 $\Bbb R$-factorizable $P$-group and deduce that there exists an $\omega $-narrow $P$-group that cannot be embedded as a subgroup into any $\Bbb R$-factorizable $P$-group. The class of $\sigma $-products of second-countable topological groups is especially interesting. We prove that {\it all subgroups\/} of the groups in this class are perfectly $\kappa $-normal, $\Bbb R$-factor\-izable, 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)\leq \omega $. (English) |
| Keyword:
|
pseudo-$\aleph_1$-compact space |
| Keyword:
|
$\Bbb R$-factorizable group |
| Keyword:
|
cellularity |
| Keyword:
|
$\sigma$-product |
| MSC:
|
22A05 |
| MSC:
|
54B50 |
| MSC:
|
54D20 |
| idZBL:
|
Zbl 1212.54099 |
| idMR:
|
MR2493947 |
| . |
| Date available:
|
2009-05-05T17:13:57Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119755 |
| . |
| Reference:
|
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