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On the complexity of subspaces of $S_{ω}$

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Uzcátegui C.

Fundamenta Mathematicae

2003

tom 176

nr 1

1-16

Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ 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 $S_ω$ is $F_{σδ}$. In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of $S_ω$. We show that $S_ω$ has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology. Moreover, a closed subset of $S_ω$ has this property iff it contains a copy of $S_ω$.

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1 Fundamenta Mathematicae

1 Uzcátegui C.

1 2003

Wyniki wyszukiwania

On the complexity of subspaces of $S_{ω}$