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Abstract

We show that a subset $X$ of a given Polish space $\mathcal X$ is $\boldsymbol{\Sigma}_{2}^{1}$ iff there is an open set $O\subseteq\mathcal X\times[\omega]^{\omega}$ such that $$ X=\{x\in\mathcal X\colon\exists r\in[\omega]^{\omega}\ \{x\}\times[r]^{\omega}\subseteq O\}. $$ This implies that if a set $U\subseteq\omega^{\omega}\times(\mathcal X\times[\omega]^{\omega})$ is universal for $G_{\delta}$ subsets of $\mathcal X\times[\omega]^{\omega}$, then the set of all $(v,x)\in\omega^{\omega}\times\mathcal X$ such that the section $U_{vx}$ has nonempty interior in the Ellentuck topology is universal for $\boldsymbol{\Sigma}_{2}^{1}$ subsets of $\mathcal X$. It follows that the $\sigma$-ideal of meager sets in the Ellentuck topology is not $\boldsymbol{\Sigma}_{2}^{1}$ on $G_{\delta}$, a fact established recently by Sabok (2012) with the help of Kleene’s Recursion Theorem.