Borel 1 type mappings and the respective equi-families
Marek Balcerzak, Ľubica Holá, Olena Karlova, Piotr Szuca
TL;DR
The paper develops a unified theory of equi-families of functions by introducing equi- variants of classical regularity properties and grounding them in an orbit map into a product space. It proves that closures under pointwise convergence preserve these equi-properties and establishes precise correspondences between equi-behavior and orbit-map properties. The authors solve two BKS problems: showing separately equi-continuous two-variable families are equi-weakly separated, and extending Grande’s result to derive Borel-class regularity of product maps from equi-α-GLP x-sections and PECP y-sections. The work yields a robust toolkit for analyzing measurability, regularity, and stability under limits for functions on product spaces, with implications for dynamical systems and analysis of two-variable maps.
Abstract
We investigate classes of functions from a topological space to a metric space that are related to those of Borel class 1. Following the idea defining an equi-Baire 1 family (due to Lecomte) we define the respective equi-families of functions from the considered classes. We observe that studying of equi-families can be reduced to the exploration of a single orbit map with values in a product space. We consider the closure of equi-families with respect to the topology of pointwise convergence. Finally, we investigate functions $f\colon X\times Y\to Z$, for metric spaces $X,Y,Z$, with sections that are equi-continuous, equi-Baire~1 or have equi-generalized Lebesgue property with respect to measurable sets of class $α$. In particular, we generalize a result of Grande.