| Title:
|
The Baire property in remainders of topological groups and other results (English) |
| Author:
|
Arhangel'skii, Alexander |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
50 |
| Issue:
|
2 |
| Year:
|
2009 |
| Pages:
|
273-279 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is established that a remainder of a non-locally compact topological group $G$ has the Baire property if and only if the space $G$ is not Čech-complete. We also show that if $G$ is a non-locally compact topological group of countable tightness, then either $G$ is submetrizable, or $G$ is the Čech-Stone remainder of an arbitrary remainder $Y$ of $G$. It follows that if $G$ and $H$ are non-submetrizable topological groups of countable tightness such that some remainders of $G$ and $H$ are homeomorphic, then the spaces $G$ and $H$ are homeomorphic. Some other corollaries and related results are presented. (English) |
| Keyword:
|
Baire property |
| Keyword:
|
$\sigma $-compact |
| Keyword:
|
Čech-complete space |
| Keyword:
|
compactification |
| Keyword:
|
Čech-Stone compactification |
| Keyword:
|
Rajkov complete |
| Keyword:
|
paracompact $p$-space |
| MSC:
|
54B05 |
| MSC:
|
54H11 |
| MSC:
|
54H15 |
| idZBL:
|
Zbl 1212.54098 |
| idMR:
|
MR2537836 |
| . |
| Date available:
|
2009-08-18T12:25:08Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133433 |
| . |
| Reference:
|
[1] Arhangel'skii A.V.: On a class of spaces containing all metric and all locally compact spaces.Mat. Sb. 67 (109) (1965), 55--88; English translation: Amer. Math. Soc. Transl. 92 (1970), 1--39. MR 0190889 |
| Reference:
|
[2] Arhangel'skii A.V.: Classes of topological groups.Russian Math. Surveys 36 (3) (1981), 151-174. MR 0622722, 10.1070/RM1981v036n03ABEH004249 |
| Reference:
|
[3] Arhangel'skii A.V.: Moscow spaces and topological groups.Topology Proc. 25 (2000), 383--416. Zbl 1027.54038, MR 1925695 |
| Reference:
|
[4] Arhangel'skii A.V.: Remainders in compactifications and generalized metrizability properties.Topology Appl. 150 (2005), 79--90. Zbl 1075.54012, MR 2133669, 10.1016/j.topol.2004.10.015 |
| Reference:
|
[5] Arhangel'skii A.V.: Two types of remainders of topological groups.Comment. Math. Univ. Carolin. 49 (2008), no. 1, 119--126. MR 2433629 |
| Reference:
|
[6] Arhangel'skii A.V., Ponomarev V.I.: Fundamentals of General Topology in Problems and Exercises.Reidel, 1984 (translated from Russian). MR 0785749 |
| Reference:
|
[7] Arhangel'skii A.V., Tkachenko M.G.: Topological Groups and Related Structures.Atlantis Press, Paris; World Scientific, Hackensack, NJ, 2008. MR 2433295 |
| Reference:
|
[8] Choban M.M.: On completions of topological groups.Vestnik Moskov. Univ. Ser. Mat. Mech. 1 (1970), 33--38 (in Russian). MR 0279226 |
| Reference:
|
[9] Choban M.M.: Topological structure of subsets of topological groups and their quotients.in Topological Structures and Algebraic Systems, Shtiintsa, Kishinev, 1977, pp. 117--163 (in Russian). |
| Reference:
|
[10] Engelking R.: General Topology.PWN, Warszawa, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[11] Rančin D.V.: Tightness, sequentiality, and closed covers.Dokl. Akad. Nauk SSSR 232 (1977), 1015--1018. MR 0436074 |
| . |
