On the closure of Baire classes under transfinite convergences

Tamás Mátrai¹

¹ Department of Analysis Eötvös Loránd University Pázmány Péter sétány 1//C 1117 Budapest, Hungary

Fundamenta Mathematicae, Tome 183 (2004) no. 2, pp. 157-168.

Résumé

Let $X$ be a Polish space and $Y$ be a separable metric space. For a fixed $\xi \omega_{1}$, consider a family $f_{\alpha}\colon\, X \to Y~( \alpha\omega_{1})$ of Baire-$\xi $ functions. Answering a question of Tomasz Natkaniec, we show that if for a function $f\colon\, X \to Y$, the set $\{ \alpha \omega_{1}\colon\, f_{\alpha}(x) \neq f(x)\}$ is finite for every $x \in X$, then $f$ itself is necessarily Baire-$\xi$. The proof is based on a characterization of $\Sigma^{0}_{\eta}$ sets which can be interesting in its own right.

Détail

DOI : 10.4064/fm183-2-6

Keywords: polish space separable metric space fixed omega consider family alpha colon alpha omega baire functions answering question tomasz natkaniec function colon set alpha omega colon alpha neq finite every itself necessarily baire proof based characterization sigma eta sets which interesting its own right