Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Fundamenta Mathematicae / All issues

Abstract

Let $X$ be a Borel subset of the Cantor set $\textbf{C}$ of additive or multiplicative class $\alpha$, and $f: X \to Y$  be a continuous function onto $Y \subset \textbf{C}$ with compact preimages of points. If the image $f(U)$ of every clopen set $U$ is the intersection of an open and a closed set, then $Y$ is a Borel set of the same class $\alpha$. This result generalizes similar results for open and closed functions.