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Abstract

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 $.