More on ordinals in topological groups

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 -