Alexander V. Arhangel'skii, Raushan Z. Buzyakova
More on ordinals in topological groups

Comment.Math.Univ.Carolin. 49,1 (2008) 127-140.

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 $\{t^2:t\in T\}$ is not closed in $H$ whenever $T\subset H$ is homeomorphic to $\omega _1$. Some other results are obtained.

Keywords: topological group, space of ordinals, $C_p(X)$
AMS Subject Classification: 54H12, 54F05