# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

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

### Tom 189 / 2006

Fundamenta Mathematicae 189 (2006), 269-284 MSC: Primary 54H05, 54H15; Secondary 54F50. DOI: 10.4064/fm189-3-4

#### Streszczenie

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.

#### Autorzy

• Todor TsankovDepartment of Mathematics 253-37
California Institute of Technology