Characterizations and Properties of Graphs of Baire Functions

TL;DR

This paper characterizes Baire-1 functions between certain topological and Banach spaces via their graphs, and extends the analysis to higher Baire classes and other spaces, providing new insights into their structural properties.

Contribution

It provides a novel characterization of Baire-1 functions through their graphs using open sets with convex vertical sections, and generalizes these results to higher Baire classes and different spaces.

Findings

Characterization of Baire-1 functions via graph intersections of open sets with convex vertical sections.

Extension of characterizations to functions of higher Baire classes and different topological spaces.

Description of the accumulation set of graphs of Baire-2 functions in certain spaces.

Abstract

Let $X$ be a paracompact topological space and $Y$ be a Banach space. In this paper, we will characterize the Baire-1 functions $f:X\rightarrow{Y}$ by their graph: namely, we will show that $f$ is a Baire-1 function if and only if its graph $gr(f)$ is the intersection of a sequence $(G_n)_{n=1}^{\infty}$ of open sets in $X\times{Y}$ such that for all $x\in{X}$ and $n\in\mathbb{N}$ the vertical section of $G_n$ is a convex set, whose diameter tends to $0$ as $n\rightarrow\infty$. Afterwards, we will discuss a similar question concerning functions of higher Baire classes and formulate some generalized results in slightly different settings: for example we require the domain to be a metrized Suslin space, while the codomain is a separable Fr\'echet space. Finally, we will characterize the accumulation set of graphs of Baire-2 functions between certain spaces.