Ideals of compact sets associated with Borel functions.

Abstract

We investigate the connection between the Borel class of a function $f$ and the Borel complexity of the set $\T(f)=\{C\in\comp(X)\colon f\rest_C\text{ is continuous}\}$ where $\comp(X)$ denotes the compact subsets of $X$ with the Hausdorff metric. For example, we show that for a function $f\colon X\to Y$ between Polish spaces; if $\T(f)$ is $F_{\sigma\delta}$ in $\comp(X)$, then $f$ is Borel class one.