More on ordinals in topological groups

Aleksander V. Arhangel'skii; Raushan Z. Buzyakova

Commentationes Mathematicae Universitatis Carolinae (2008)

  • Volume: 49, Issue: 1, page 127-140
  • ISSN: 0010-2628

Abstract

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Let τ be an uncountable regular cardinal and G a T 1 topological group. We prove the following statements: (1) If τ 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 τ × τ embeds in G as a closed subspace. (2) If G is Abelian, algebraically generated by τ G , and the order of every element does not exceed 3 then τ × τ is not embeddable in G . (3) There exists an Abelian topological group H such that ω 1 is homeomorphic to a closed subspace of H and { t 2 : t T } is not closed in H whenever T H is homeomorphic to ω 1 . Some other results are obtained.

How to cite

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Arhangel'skii, Aleksander V., and Buzyakova, Raushan Z.. "More on ordinals in topological groups." Commentationes Mathematicae Universitatis Carolinae 49.1 (2008): 127-140. <http://eudml.org/doc/250290>.

@article{Arhangelskii2008,
abstract = {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 $\lbrace t^2:t\in T\rbrace $ is not closed in $H$ whenever $T\subset H$ is homeomorphic to $\omega _1$. Some other results are obtained.},
author = {Arhangel'skii, Aleksander V., Buzyakova, Raushan Z.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {topological group; space of ordinals; $C_p(X)$; space of ordinals; },
language = {eng},
number = {1},
pages = {127-140},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {More on ordinals in topological groups},
url = {http://eudml.org/doc/250290},
volume = {49},
year = {2008},
}

TY - JOUR
AU - Arhangel'skii, Aleksander V.
AU - Buzyakova, Raushan Z.
TI - More on ordinals in topological groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2008
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 49
IS - 1
SP - 127
EP - 140
AB - 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 $\lbrace t^2:t\in T\rbrace $ is not closed in $H$ whenever $T\subset H$ is homeomorphic to $\omega _1$. Some other results are obtained.
LA - eng
KW - topological group; space of ordinals; $C_p(X)$; space of ordinals;
UR - http://eudml.org/doc/250290
ER -

References

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  1. Arhangelskii A., Topological function spaces, Math. Appl., vol. 78, Kluwer Academic Publisher, Dordrecht, 1992. MR1144519
  2. Buzyakova R.Z., 10.4064/fm196-2-3, Fund. Math. 196 (2007), 127-138. (2007) Zbl1133.54022MR2342623DOI10.4064/fm196-2-3
  3. Comfort W.W., Ross K.A., 10.2140/pjm.1966.16.483, Pacific J. Math. 16 (1966), 483-496. (1966) Zbl0214.28502MR0207886DOI10.2140/pjm.1966.16.483
  4. Engelking R., General Topology, Sigma Series in Pure Mathematics, 6, Heldermann, Berlin, revised ed., 1989. Zbl0684.54001MR1039321
  5. Kunen K., Set Theory, Elsevier, 1980. Zbl0960.03033MR0597342
  6. Pontryagin L.S., Continuous Groups, Moscow; English translation: {Topological Groups}, Princeton University Press, Princeton, 1939. Zbl0659.22001

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