Characterizations and Properties of Graphs of Baire Functions
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arXiv
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| Format: | Preprint |
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2016
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| _version_ | 1866914579606405120 |
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| author | Maga, Balázs |
| author_facet | Maga, Balázs |
| contents | 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 Fréchet space. Finally, we will characterize the accumulation set of graphs of Baire-2 functions between certain spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1611_00678 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Characterizations and Properties of Graphs of Baire Functions Maga, Balázs Classical Analysis and ODEs Functional Analysis General Topology 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 Fréchet space. Finally, we will characterize the accumulation set of graphs of Baire-2 functions between certain spaces. |
| title | Characterizations and Properties of Graphs of Baire Functions |
| topic | Classical Analysis and ODEs Functional Analysis General Topology |
| url | https://arxiv.org/abs/1611.00678 |