Characterizations and Properties of Graphs of Baire Functions

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.