Preservation of the Borel class under open-$LC$ functions

TL;DR

This paper proves that under certain conditions, the Borel class of a set is preserved when applying an open-$LC$ function, extending previous results for open and closed functions.

Contribution

It introduces a generalization showing Borel class preservation under open-$LC$ functions with specific properties, broadening the understanding of Borel set transformations.

Findings

Borel class is preserved under the specified open-$LC$ functions.

The result generalizes earlier theorems for open and closed functions.

The image of a Borel set remains Borel of the same class under these functions.

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 with compact preimages of points onto $Y \subset \textbf{C}.$ 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. This result generalizes similar results for open and closed functions.