Ordinals in topological groups

Raushan Z. Buzyakova¹

¹ Mathematics Department UNCG Greensboro, NC 27402, U.S.A.

Fundamenta Mathematicae, Tome 196 (2007) no. 2, pp. 127-138

Résumé

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 $G$.

Détail

DOI : 10.4064/fm196-2-3

Keywords: uncountable regular cardinal tau tau embed topological group closed subspaces normal prove uncountable regular cardinal cannot embedded torsion abelian group hereditarily normal these results corollaries main results about ordinals topological groups state main results tau uncountable regular cardinal topological group prove among others following statements tau tau embed closedly tau times tau embeds closedly nbsp tau embeds nbsp abelian order every non neutral element greater prod tau embeds nbsp previous statement holds tau replaced tau nbsp abelian algebraically generated tau subset order every element does exceed n prod tau embeddable nbsp