| Title:
|
More on ordinals in topological groups (English) |
| Author:
|
Arhangel'skii, Aleksander V. |
| Author:
|
Buzyakova, Raushan Z. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
49 |
| Issue:
|
1 |
| Year:
|
2008 |
| Pages:
|
127-140 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\tau$ be an uncountable regular cardinal and $G$ a $T_1$ topological group. We prove the following statements: (1) If $\tau$ is homeomorphic to a closed subspace of $G$, $G$ is Abelian, and the order of every non-neutral element of $G$ is greater than $5$ then $\tau\times\tau$ embeds in $G$ as a closed subspace. (2) If $G$ is Abelian, algebraically generated by $\tau\subset G$, and the order of every element does not exceed $3$ then $\tau\times \tau$ is not embeddable in $G$. (3) There exists an Abelian topological group $H$ such that $\omega_1$ is homeomorphic to a closed subspace of $H$ and $\{t^2:t\in T\}$ is not closed in $H$ whenever $T\subset H$ is homeomorphic to $\omega_1$. Some other results are obtained. (English) |
| Keyword:
|
topological group |
| Keyword:
|
space of ordinals |
| Keyword:
|
$C_p(X)$ |
| MSC:
|
54F05 |
| MSC:
|
54H12 |
| idZBL:
|
Zbl 1212.54103 |
| idMR:
|
MR2433630 |
| . |
| Date available:
|
2009-05-05T17:06:58Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119707 |
| . |
| Reference:
|
[ARH] Arhangelskii A.: Topological function spaces.Math. Appl., vol. 78, Kluwer Academic Publisher, Dordrecht, 1992. MR 1144519 |
| Reference:
|
[BUZ] Buzyakova R.Z.: Ordinals in topological groups.Fund. Math. 196 (2007), 127-138. Zbl 1133.54022, MR 2342623, 10.4064/fm196-2-3 |
| Reference:
|
[C&R] 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, 10.2140/pjm.1966.16.483 |
| Reference:
|
[ENG] Engelking R.: General Topology.Sigma Series in Pure Mathematics, 6, Heldermann, Berlin, revised ed., 1989. Zbl 0684.54001, MR 1039321 |
| Reference:
|
[KUN] Kunen K.: Set Theory.Elsevier, 1980. Zbl 0960.03033, MR 0597342 |
| Reference:
|
[PON] Pontryagin L.S.: Continuous Groups.Moscow; English translation: {Topological Groups}, Princeton University Press, Princeton, 1939. Zbl 0659.22001 |
| . |
