| Title:
|
Two types of remainders of topological groups (English) |
| Author:
|
Arhangel'skii, A. V. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
49 |
| Issue:
|
1 |
| Year:
|
2008 |
| Pages:
|
119-126 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We prove a Dichotomy Theorem: for each Hausdorff compactification $bG$ of an arbitrary topological group $G$, the remainder $bG\setminus G$ is either pseudocompact or Lindelöf. It follows that if a remainder of a topological group is paracompact or Dieudonne complete, then the remainder is Lindelöf, and the group is a paracompact $p$-space. This answers a question in A.V. Arhangel'skii, {\it Some connections between properties of topological groups and of their remainders\/}, Moscow Univ. Math. Bull. 54:3 (1999), 1--6. It is shown that every Tychonoff space can be embedded as a closed subspace in a pseudocompact remainder of some topological group. We also establish some other results and present some examples and questions. (English) |
| Keyword:
|
remainder |
| Keyword:
|
compactification |
| Keyword:
|
topological group |
| Keyword:
|
$p$-space |
| Keyword:
|
Lindelöf $p$-space |
| Keyword:
|
metrizability |
| Keyword:
|
countable type |
| Keyword:
|
Lindelöf space |
| Keyword:
|
pseudocompact space |
| Keyword:
|
$\pi $-base |
| Keyword:
|
compactification |
| MSC:
|
54A25 |
| MSC:
|
54B05 |
| idZBL:
|
Zbl 1212.54086 |
| idMR:
|
MR2433629 |
| . |
| Date available:
|
2009-05-05T17:06:53Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119706 |
| . |
| 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.: Some connections between properties of topological groups and of their remainders.Moscow Univ. Math. Bull. 54:3 (1999), 1-6. MR 1711899 |
| Reference:
|
[4] Arhangel'skii A.V.: Topological invariants in algebraic environment.in: Recent Progress in General Topology 2, eds. M. Hušek, Jan van Mill, North-Holland, Amsterdam, 2002, pp.1-57. Zbl 1030.54026, MR 1969992 |
| Reference:
|
[5] 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:
|
[6] Arhangel'skii A.V.: More on remainders close to metrizable spaces.Topology Appl. 154 (2007), 1084-1088. Zbl 1144.54001, MR 2298623, 10.1016/j.topol.2006.10.008 |
| Reference:
|
[7] Engelking R.: General Topology.PWN, Warszawa, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[8] Filippov V.V.: On perfect images of paracompact $p$-spaces.Soviet Math. Dokl. 176 (1967), 533-536. MR 0222853 |
| Reference:
|
[9] Henriksen M., Isbell J.R.: Some properties of compactifications.Duke Math. J. 25 (1958), 83-106. Zbl 0081.38604, MR 0096196, 10.1215/S0012-7094-58-02509-2 |
| Reference:
|
[10] Tkachenko M.G.: The Suslin property in free topological groups over compact spaces (Russian).Mat. Zametki 34 (1983), 601-607; English translation: Math. Notes 34 (1983), 790-793. MR 0722229 |
| Reference:
|
[11] Roelke W., Dierolf S.: Uniform Structures on Topological Groups and their Quotients.McGraw-Hill, New York, 1981. |
| . |
