| Title:
|
Subgroups of $\Bbb R$-factorizable groups (English) |
| Author:
|
Hernández, Constancio |
| Author:
|
Tkačenko, Michael |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
39 |
| Issue:
|
2 |
| Year:
|
1998 |
| Pages:
|
371-378 |
| . |
| Category:
|
math |
| . |
| Summary:
|
The properties of $\Bbb R$-factorizable groups and their subgroups are studied. We show that a locally compact group $G$ is $\Bbb R$-factorizable if and only if $G$ is $\sigma$-compact. It is proved that a subgroup $H$ of an $\Bbb R$-factorizable group $G$ is $\Bbb R$-factorizable if and only if $H$ is $z$-embedded in $G$. Therefore, a subgroup of an $\Bbb R$-factorizable group need not be $\Bbb R$-factorizable, and we present a method for constructing non-$\Bbb R$-factorizable dense subgroups of a special class of $\Bbb R$-factorizable groups. Finally, we construct a closed $G_{\delta}$-subgroup of an $\Bbb R$-fac\-torizable group which is not $\Bbb R$-factorizable. (English) |
| Keyword:
|
$\Bbb R$-factorizable group |
| Keyword:
|
$z$-embedded set |
| Keyword:
|
$\aleph_0$-bounded group |
| Keyword:
|
$P$-group |
| Keyword:
|
Lindelöf group |
| MSC:
|
22A05 |
| MSC:
|
22D05 |
| MSC:
|
54C50 |
| MSC:
|
54H11 |
| idZBL:
|
Zbl 1100.54026 |
| idMR:
|
MR1651979 |
| . |
| Date available:
|
2009-01-08T18:41:12Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119014 |
| . |
| Reference:
|
[1] Comfort W.W.: Compactness like properties for generalized weak topological sums.Pacific J. Math. 60 (1975), 31-37. Zbl 0307.54016, MR 0431088 |
| Reference:
|
[2] Comfort W.W., Ross K.A.: Pseudocompactness and uniform continuity in topological groups.Pacific J. Math. 16 (1966), 483-496. Zbl 0214.28502, MR 0207886 |
| Reference:
|
[3] Guran I.I.: On topological groups close to being Lindelöf.Soviet Math. Dokl. 23 (1981), 173-175. Zbl 0478.22002 |
| Reference:
|
[4] Hernández S., Sanchiz M., Tkačenko M.: Bounded sets in spaces and topological groups.submitted for publication. |
| Reference:
|
[5] Engelking R.: General Topology.Heldermann Verlag, 1989. Zbl 0684.54001, MR 1039321 |
| Reference:
|
[6] Pontryagin L.S.: Continuous Groups.Princeton Univ. Press, Princeton, 1939. Zbl 0659.22001 |
| Reference:
|
[7] Tkačenko M.G.: Subgroups, quotient groups and products of $\Bbb R$-factorizable groups.Topology Proceedings 16 (1991), 201-231. MR 1206464 |
| Reference:
|
[8] Tkačenko M.G.: Factorization theorems for topological groups and their applications.Topology Appl. 38 (1991), 21-37. MR 1093863 |
| . |
