Baire property of space of Baire-one functions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866909374728896512 |
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| author | Osipov, Alexander V. |
| author_facet | Osipov, Alexander V. |
| contents | A topological space $X$ is Baire if the Baire Category Theorem holds for $X$, i.e., the intersection of any sequence of open dense subsets of $X$ is dense in $X$. One of the interesting problems for the space $B_1(X)$ of all Baire-one real-valued functions is characterization topological space $X$ for which the function space $B_1(X)$ is Baire. In this paper, we solve this problem, namely, we have obtained a characterization when a function space $B_1(X)$ has the Baire property for any Tychonoff space $X$. Also we proved that $B_1(X)$ is Baire for any $γ$-space $X$. This answers a question posed recently by T. Banakh and S. Gabriyelyan. We also conclude that, it is consistent there are no uncountable separable metrizable space $X$ such that $B_1(X)$ is countable dense homogeneous. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_15496 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Baire property of space of Baire-one functions Osipov, Alexander V. General Topology A topological space $X$ is Baire if the Baire Category Theorem holds for $X$, i.e., the intersection of any sequence of open dense subsets of $X$ is dense in $X$. One of the interesting problems for the space $B_1(X)$ of all Baire-one real-valued functions is characterization topological space $X$ for which the function space $B_1(X)$ is Baire. In this paper, we solve this problem, namely, we have obtained a characterization when a function space $B_1(X)$ has the Baire property for any Tychonoff space $X$. Also we proved that $B_1(X)$ is Baire for any $γ$-space $X$. This answers a question posed recently by T. Banakh and S. Gabriyelyan. We also conclude that, it is consistent there are no uncountable separable metrizable space $X$ such that $B_1(X)$ is countable dense homogeneous. |
| title | Baire property of space of Baire-one functions |
| topic | General Topology |
| url | https://arxiv.org/abs/2110.15496 |