Completeness and reflexivity type properties of $B_1(X)$
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866914244360929280 |
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| author | Gabriyelyan, Saak Osipov, Alexander V. Reznichenko, Evgenii |
| author_facet | Gabriyelyan, Saak Osipov, Alexander V. Reznichenko, Evgenii |
| contents | For a Tychonoff space $X$, $B_1(X)$ denotes the space of all Baire-one functions on $X$ endowed with the pointwise topology. We prove that the following assertions are equivalent: (1) $B_1(X)$ is a (semi-)Montel space, (2) $B_1(X)$ is a (semi-)reflexive space, (3) $B_1(X)$ is a (quasi-)complete space, (4) $B_1(X)=\mathbb{R}^X$, (5) $X$ is a $Q_f$-space. It is proved that $B_1(X)$ is sequentially complete iff $B_1(X)$ is locally complete iff $X$ is a $CZ$-space. In the case when $K$ is a compact space, we show that $B_1(K)$ is locally complete iff $K$ is scattered. We thoroughly study the case when $X$ is a separable metrizable space. Numerous distinguished examples are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00733 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Completeness and reflexivity type properties of $B_1(X)$ Gabriyelyan, Saak Osipov, Alexander V. Reznichenko, Evgenii General Topology Functional Analysis For a Tychonoff space $X$, $B_1(X)$ denotes the space of all Baire-one functions on $X$ endowed with the pointwise topology. We prove that the following assertions are equivalent: (1) $B_1(X)$ is a (semi-)Montel space, (2) $B_1(X)$ is a (semi-)reflexive space, (3) $B_1(X)$ is a (quasi-)complete space, (4) $B_1(X)=\mathbb{R}^X$, (5) $X$ is a $Q_f$-space. It is proved that $B_1(X)$ is sequentially complete iff $B_1(X)$ is locally complete iff $X$ is a $CZ$-space. In the case when $K$ is a compact space, we show that $B_1(K)$ is locally complete iff $K$ is scattered. We thoroughly study the case when $X$ is a separable metrizable space. Numerous distinguished examples are given. |
| title | Completeness and reflexivity type properties of $B_1(X)$ |
| topic | General Topology Functional Analysis |
| url | https://arxiv.org/abs/2601.00733 |