Open--constructible functions

TL;DR

This paper characterizes functions that map open sets into Boolean algebra elements generated by open and closed sets, showing they can be decomposed into parts where the function is open.

Contribution

It provides a new characterization of continuous functions with specific Boolean algebra properties, demonstrating their decomposition into open functions on subsets.

Findings

Functions with certain Boolean algebra properties can be decomposed into open functions on subsets.

Such functions cover their entire image through these open subsets.

The result links Boolean algebra conditions to the topological openness of 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).$