Continuous images of Lindelöf p -groups, σ -compact groups, and related results

Aleksander V. Arhangel'skii

Commentationes Mathematicae Universitatis Carolinae (2019)

  • Volume: 60, Issue: 4, page 463-471
  • ISSN: 0010-2628

Abstract

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It is shown that there exists a σ -compact topological group which cannot be represented as a continuous image of a Lindelöf p -group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf p -groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space Y is a continuous image of a Lindelöf p -group, then there exists a covering γ of Y by dyadic compacta such that | γ | ≤ 2 ω . We also show that if a homogeneous compact space Y is a continuous image of a c d c -group G , then Y is a dyadic compactum (Corollary 3.11).

How to cite

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Arhangel'skii, Aleksander V.. "Continuous images of Lindelöf $p$-groups, $\sigma $-compact groups, and related results." Commentationes Mathematicae Universitatis Carolinae 60.4 (2019): 463-471. <http://eudml.org/doc/295070>.

@article{Arhangelskii2019,
abstract = {It is shown that there exists a $\sigma $-compact topological group which cannot be represented as a continuous image of a Lindelöf $p$-group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf $p$-groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space $Y$ is a continuous image of a Lindelöf $p$-group, then there exists a covering $\gamma $ of $Y$ by dyadic compacta such that $|\gamma |\le 2^\omega $. We also show that if a homogeneous compact space $Y$ is a continuous image of a $cdc$-group $G$, then $Y$ is a dyadic compactum (Corollary 3.11).},
author = {Arhangel'skii, Aleksander V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Lindelöf $p$-group; homogeneous space; Lindelöf $\Sigma $-space; dyadic compactum; countable tightness; $\sigma $-compact; $cdc$-group; $p$-space},
language = {eng},
number = {4},
pages = {463-471},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Continuous images of Lindelöf $p$-groups, $\sigma $-compact groups, and related results},
url = {http://eudml.org/doc/295070},
volume = {60},
year = {2019},
}

TY - JOUR
AU - Arhangel'skii, Aleksander V.
TI - Continuous images of Lindelöf $p$-groups, $\sigma $-compact groups, and related results
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2019
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 60
IS - 4
SP - 463
EP - 471
AB - It is shown that there exists a $\sigma $-compact topological group which cannot be represented as a continuous image of a Lindelöf $p$-group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf $p$-groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space $Y$ is a continuous image of a Lindelöf $p$-group, then there exists a covering $\gamma $ of $Y$ by dyadic compacta such that $|\gamma |\le 2^\omega $. We also show that if a homogeneous compact space $Y$ is a continuous image of a $cdc$-group $G$, then $Y$ is a dyadic compactum (Corollary 3.11).
LA - eng
KW - Lindelöf $p$-group; homogeneous space; Lindelöf $\Sigma $-space; dyadic compactum; countable tightness; $\sigma $-compact; $cdc$-group; $p$-space
UR - http://eudml.org/doc/295070
ER -

References

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  5. Arhangel'skii A. V., van Mill J., Topological homogeneity, Recent Progress in General Topology, III, Atlantis Press, Paris, 2014, 1–68. MR3204728
  6. Arhangel'skiĭ A. V., Ponomarev V. I., Dyadic bicompacta, Dokl. Akad. Nauk SSSR 182 (1968), 993–996 (Russian); English translation in Soviet Math. Dokl. 9 (1968), 1220–1224. MR0235518
  7. Arhangel'skii A., Tkachenko M., Topological Groups and Related Structures, Atlantis Studies in Mathematics, 1, Atlantis Press, Paris, World Scientific Publishing, Hackensack, 2008. MR2433295
  8. Čoban M. M., Topological structure of subsets of topological groups and their quotient spaces, Topological Structures and Algebraic Systems, Mat. Issled. Vyp. 44 (1977), 117–163, 181 (Russian). MR0492040
  9. Engelking R., General Topology, Mathematical Monographs, 60, PWN—Polish Scientific Publishers, Warsaw, 1977. Zbl0684.54001MR0500780
  10. Nagami K., Σ -spaces, Fund. Math. 65 (1969), 169–192. Zbl0181.50701MR0257963
  11. Rančin D. V., Tightness, sequentiality, and closed coverings, Dokl. Akad. Nauk SSSR 232 (1977), no. 5, 1015–1018 (Russian); English translation in Soviet Math. Dokl. 18 (1977), no. 1, 196–200. MR0436074

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