Ordinals in topological groups
Tom 196 / 2007
Streszczenie
We show that if an uncountable regular cardinal $\tau$ and $\tau+1$ embed in a topological group $G$ as closed subspaces then $G$ is not normal. We also prove that an uncountable regular cardinal cannot be embedded in a torsion free Abelian group that is hereditarily normal. These results are corollaries to our main results about ordinals in topological groups. To state the main results, let $\tau$ be an uncountable regular cardinal and $G$ a $T_1$ topological group. We prove, among others, the following statements: (1) If $\tau$ and $\tau+1$ embed closedly in $G$ then $\tau\times (\tau +1)$ embeds closedly in $G$; (2) If $\tau$ embeds in $G$, $G$ is Abelian, and the order of every non-neutral element of $G$ is greater than $2^N -1$ then $\prod_{i\in N}\tau$ embeds in $G$; (3) The previous statement holds if $\tau$ is replaced by $\tau + 1$; (4) If $G$ is Abelian, algebraically generated by $\tau +1\subset G$, and the order of every element does not exceed $2^N-1$ then $\prod_{i\in N}(\tau +1 )$ is not embeddable in $G$.
