Compactifications of $\mathbb{N}$ and Polishable subgroups of $S_{\infty}$

Todor Tsankov¹

¹ Department of Mathematics 253-37 California Institute of Technology Pasadena, CA 91125, U.S.A.

Fundamenta Mathematicae, Tome 189 (2006) no. 3, pp. 269-284

Résumé

We study homeomorphism groups of metrizable compactifications of $\mathbb{N}$. All of those groups can be represented as almost zero-dimensional Polishable subgroups of the group $S_\infty$. As a corollary, we show that all Polish groups are continuous homomorphic images of almost zero-dimensional Polishable subgroups of $S_\infty$. We prove a sufficient condition for these groups to be one-dimensional and also study their descriptive complexity. In the last section we associate with every Polishable ideal on $\mathbb{N}$ a certain Polishable subgroup of $S_\infty$ which shares its topological dimension and descriptive complexity.

Détail

DOI : 10.4064/fm189-3-4

Keywords: study homeomorphism groups metrizable compactifications mathbb those groups represented almost zero dimensional polishable subgroups group infty corollary polish groups continuous homomorphic images almost zero dimensional polishable subgroups infty prove sufficient condition these groups one dimensional study their descriptive complexity section associate every polishable ideal mathbb certain polishable subgroup infty which shares its topological dimension descriptive complexity