| Title:
|
Continuous images of Lindelöf $p$-groups, $\sigma $-compact groups, and related results (English) |
| Author:
|
Arhangel'skii, Alexander V. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
60 |
| Issue:
|
4 |
| Year:
|
2019 |
| Pages:
|
463-471 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 |\leq 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). (English) |
| Keyword:
|
Lindelöf $p$-group |
| Keyword:
|
homogeneous space |
| Keyword:
|
Lindelöf $\Sigma $-space |
| Keyword:
|
dyadic compactum |
| Keyword:
|
countable tightness |
| Keyword:
|
$\sigma $-compact |
| Keyword:
|
$cdc$-group |
| Keyword:
|
$p$-space |
| MSC:
|
54A25 |
| MSC:
|
54B05 |
| idZBL:
|
Zbl 07177883 |
| idMR:
|
MR4061356 |
| DOI:
|
10.14712/1213-7243.2019.027 |
| . |
| Date available:
|
2020-02-10T16:48:32Z |
| Last updated:
|
2022-01-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147979 |
| . |
| Reference:
|
[1] Arhangel'skiĭ A. V.: A class of spaces which contains all metric and all locally compact spaces.Mat. Sb. (N.S.) 67(109) (1965), 55–88 (Russian); English translation in Amer. Math. Soc. Transl. 92 (1970), 1–39. MR 0190889 |
| Reference:
|
[2] Arhangel'skii A. V.: The power of bicompacta with first axiom of countability.Dokl. Akad. Nauk SSSR 187 (1969), 967–970 (Russian); English translation in Soviet Math. Dokl. 10 (1969), 951–955. MR 0251695 |
| Reference:
|
[3] Arhangel'skiĭ A. V.: Topological homogeneity. Topological groups and their continuous images.Uspekhi Mat. Nauk 42 (1987), no. 2(254), 69–105 (Russian); English translation in Russian Math. Surveys 42 (1987), 83–131. MR 0898622 |
| Reference:
|
[4] Arhangel'skiĭ A. V.: Topological Function Spaces.Mathematics and Its Applications (Soviet Series), 78, Kluwer Academic Publishers Group, Dordrecht, 1992. MR 1144519, 10.1007/978-94-011-2598-7_4 |
| Reference:
|
[5] Arhangel'skii A. V., van Mill J.: Topological homogeneity.Recent Progress in General Topology, III, Atlantis Press, Paris, 2014, 1–68. MR 3204728 |
| Reference:
|
[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. MR 0235518 |
| Reference:
|
[7] Arhangel'skii A., Tkachenko M.: Topological Groups and Related Structures.Atlantis Studies in Mathematics, 1, Atlantis Press, Paris, World Scientific Publishing, Hackensack, 2008. MR 2433295 |
| Reference:
|
[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). MR 0492040 |
| Reference:
|
[9] Engelking R.: General Topology.Mathematical Monographs, 60, PWN—Polish Scientific Publishers, Warsaw, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[10] Nagami K.: $\Sigma $-spaces.Fund. Math. 65 (1969), 169–192. Zbl 0181.50701, MR 0257963 |
| Reference:
|
[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. MR 0436074 |
| . |
