On Baire property, compactness and completeness properties of spaces of Baire functions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866910858750197760 |
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| author | Osipov, Alexander V. |
| author_facet | Osipov, Alexander V. |
| contents | A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. One of the interesting problems for the space of Baire functions is the Banakh-Gabriyelyan problem: Let $α$ be a countable ordinal. Characterize topological spaces $X$ and $Y$ for which the function space $B_α(X,Y)$ is Baire. In this paper, for any Frechet space $Y$ , we have obtained a characterization topological spaces $X$ for which the function space $B_α(X,Y)$ is Baire. In particular, we proved that $B_α(X,\mathbb{R})$ is Baire if and only if $B_α(X,Y)$ is Baire for any Banach space $Y$. Also we proved that many completeness and compactness properties coincide in spaces $B_α(X,Y)$ for any Frechet space $Y$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_10765 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On Baire property, compactness and completeness properties of spaces of Baire functions Osipov, Alexander V. General Topology A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. One of the interesting problems for the space of Baire functions is the Banakh-Gabriyelyan problem: Let $α$ be a countable ordinal. Characterize topological spaces $X$ and $Y$ for which the function space $B_α(X,Y)$ is Baire. In this paper, for any Frechet space $Y$ , we have obtained a characterization topological spaces $X$ for which the function space $B_α(X,Y)$ is Baire. In particular, we proved that $B_α(X,\mathbb{R})$ is Baire if and only if $B_α(X,Y)$ is Baire for any Banach space $Y$. Also we proved that many completeness and compactness properties coincide in spaces $B_α(X,Y)$ for any Frechet space $Y$. |
| title | On Baire property, compactness and completeness properties of spaces of Baire functions |
| topic | General Topology |
| url | https://arxiv.org/abs/2211.10765 |