A characterization of $\boldsymbol {\Sigma }_{2}^{1}$ sets

Janusz Pawlikowski¹

¹ Department of Mathematics University of Wrocław Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland

Fundamenta Mathematicae, Tome 236 (2017) no. 1, pp. 45-49.

Résumé

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.

Détail

DOI : 10.4064/fm61-4-2016

Keywords: subset given polish space mathcal boldsymbol sigma there set subseteq mathcal times omega omega mathcal colon exists omega omega times omega subseteq implies set subseteq omega omega times mathcal times omega omega universal delta subsets mathcal times omega omega set omega omega times mathcal section has nonempty interior ellentuck topology universal boldsymbol sigma subsets mathcal follows nbsp sigma ideal meager sets ellentuck topology boldsymbol sigma delta established recently sabok help kleene recursion theorem