# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## On the complexity of subspaces of $S_{\omega}$

### Tom 176 / 2003

Fundamenta Mathematicae 176 (2003), 1-16 MSC: Primary 54H05, 03E15; Secondary 54D55. DOI: 10.4064/fm176-1-1

#### Streszczenie

Let $(X,\tau )$ be a countable topological space. We say that $\tau$ is an analytic (resp. Borel) topology if $\tau$ as a subset of the Cantor set $2^X$ (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel'skiĭ–Franklin space $S_\omega$ is $F_{\sigma \delta }$. In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of $S_\omega$. We show that $S_\omega$ has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology. Moreover, a closed subset of $S_\omega$ has this property iff it contains a copy of $S_\omega$.

#### Autorzy

• Carlos UzcáteguiDepartamento de Matemáticas