# The Lindelöf property and pseudo-${\aleph}_{1}$-compactness in spaces and topological groups

Constancio Hernández; Mihail G. Tkachenko

Commentationes Mathematicae Universitatis Carolinae (2008)

- Volume: 49, Issue: 4, page 677-692
- ISSN: 0010-2628

## Access Full Article

top## Abstract

top## How to cite

topHernández, Constancio, and Tkachenko, Mihail G.. "The Lindelöf property and pseudo-$\aleph _1$-compactness in spaces and topological groups." Commentationes Mathematicae Universitatis Carolinae 49.4 (2008): 677-692. <http://eudml.org/doc/250494>.

@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 -

## References

top- Arhangel'skii A.V., Ranchin D.V., Everywhere dense subspaces of topological products and properties associated with final compactness, Vestnik Moskov. Univ. Ser. I Mat. Meh. (1982), 6 21-28. (1982) MR0685258
- Blair R.L., 10.4153/CJM-1976-068-9, Canad. J. Math. 28 (1976), 673-690. (1976) Zbl0359.54009MR0420542DOI10.4153/CJM-1976-068-9
- Comfort W.W., 10.2140/pjm.1975.60.31, Pacific J. Math. 60 (1975), 31-37. (1975) Zbl0307.54016MR0431088DOI10.2140/pjm.1975.60.31
- Comfort W.W., Negrepontis S., The Theory of Ultrafilters, Springer, New York, Heidelberg, Berlin, 1974. Zbl0298.02004MR0396267
- de Leo L., Tkachenko M.G., The maximal $\xf8mega$-narrow group topology on Abelian groups, Houston J. Math., to appear.
- Efimov B.A., Dyadic compacta, Trudy Moskov. Mat. Obshch. 14 (1965), 211-247 (in Russian). (1965) MR0202105
- Frolík Z., Generalizations of compactness and the Lindelöf property, Czechoslovak Math. J. 9 (84) (1959), 172-211. (1959) MR0105075
- Frolík Z., The topological product of two pseudocompact spaces, Czechoslovak Math. J. 10 (1960), 339-349. (1960) MR0116304
- Glicksberg I., Stone-Čech compactifications of products, Trans. Amer. Math. Soc. 90 (1959), 369-382. (1959) Zbl0089.38702MR0105667
- Guran I., On topological groups close to being Lindelöf, Soviet Math. Dokl. 23 (1981), 173-175. (1981) Zbl0478.22002
- Hernández C., Tkachenko M.G., Subgroups and products of $\mathbb{R}$-factorizable groups, Comment. Math. Univ. Carolin. 45 1 (2004), 153-167. (2004) MR2076867
- Jech T., Set Theory, Springer Monographs in Mathematics, Springer, Berlin, 2003. Zbl1007.03002MR1940513
- Kombarov A.P., Malykhin V.I., On $\Sigma $-products, Dokl. Akad. Nauk SSSR 213 4 (1973), 774-776 (in Russian). (1973) MR0339073
- Noble N., 10.1090/S0002-9947-1969-0250261-4, Trans. Amer. Math. Soc. 140 (1969), 381-391. (1969) Zbl0192.59701MR0250261DOI10.1090/S0002-9947-1969-0250261-4
- Schepin E.V., Real-valued functions and canonical sets in Tychonoff products and topological groups, Russian Math. Surveys 31 (1976), 19-30. (1976)
- Tkachenko M.G., Some results on inverse spectra I, Comment. Math. Univ. Carolin. 22 3 (1981), 621-633. (1981) Zbl0478.54005MR0633589
- Tkachenko M.G., 10.1016/S0166-8641(98)00051-0, Topology Appl. 86 (1998), 179-231. (1998) Zbl0955.54013MR1623960DOI10.1016/S0166-8641(98)00051-0
- Tkachenko M.G., Complete ${\aleph}_{0}$-bounded groups need not be $\mathbb{R}$-factorizable, Comment. Math. Univ. Carolin. 42 3 (2001), 551-559. (2001) Zbl1053.54045MR1860244
- Tkachenko M.G., 10.1016/S0166-8641(03)00217-7, Topology Appl. 136 (2004), 135-167. (2004) Zbl1039.54020MR2023415DOI10.1016/S0166-8641(03)00217-7
- Uspenskij V.V., On continuous images of Lindelöf topological groups, Soviet Math. Dokl. 32 (1985), 802-806. (1985) Zbl0602.22003MR0821360
- Uspenskij V.V., Topological groups and Dugundji compacta, Math. USSR Sbornik 67 (1990), 555-580; Russian original in Mat. Sbornik 180 (1989), 1092-1118. (1990) Zbl0702.22002MR1019483

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.