Open--constructible functions

Abstract

We prove that if a continuous function $f:X\to f(X)$ takes open sets into elements of the Boolean algebra generated by open and closed subsets in $f(X)$, then there exist $X_n\subset X,$ $(n\in \omega )$ such that $f$ is open on every $X_n$ and $f(X_n)$ cover $f(X).$