Functions on products $X \times Y$ with applications to Ascoli spaces, $k_{\mathbb{R}}$-spaces and $s_{\mathbb{R}}$-spaces
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Gabriyelyan, Saak Reznichenko, Evgenii |
| author_facet | Gabriyelyan, Saak Reznichenko, Evgenii |
| contents | We prove that a Tychonoff space $X$ is (sequentially) Ascoli iff for every compact space $K$ (resp., for a convergent sequence $\mathbf{s}$), each separately continuous $k$-continuous function $Φ:X\times K\to \mathbb{R}$ is continuous. We apply these characterizations to show that an open subspace of a (sequentially) Ascoli space is (sequentially) Ascoli, and that the $μ$-completion and the Dieudonné completion of a (sequentially) Ascoli space are (sequentially) Ascoli. We give also cover-type characterizations of Ascoli spaces and suggest an easy method of construction of pseudocompact Ascoli spaces which are not $k_\mathbb{R}$-spaces and show that each space $X$ can be closely embedded into such a space. Using a different method we prove Hušek's theorem: a Tychonoff space $Y$ is a locally pseudocompact $k_\mathbb{R}$-space iff $X\times Y$ is a $k_\mathbb{R}$-space for each $k_\mathbb{R}$-space $X$. It is proved that $X$ is an $s_\mathbb{R}$-space iff for every locally compact sequential space $K$, each $s$-continuous function $f:X\times K\to\mathbb{R}$ is continuous. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09670 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Functions on products $X \times Y$ with applications to Ascoli spaces, $k_{\mathbb{R}}$-spaces and $s_{\mathbb{R}}$-spaces Gabriyelyan, Saak Reznichenko, Evgenii General Topology 54A05 54B05 54B15 54D60 We prove that a Tychonoff space $X$ is (sequentially) Ascoli iff for every compact space $K$ (resp., for a convergent sequence $\mathbf{s}$), each separately continuous $k$-continuous function $Φ:X\times K\to \mathbb{R}$ is continuous. We apply these characterizations to show that an open subspace of a (sequentially) Ascoli space is (sequentially) Ascoli, and that the $μ$-completion and the Dieudonné completion of a (sequentially) Ascoli space are (sequentially) Ascoli. We give also cover-type characterizations of Ascoli spaces and suggest an easy method of construction of pseudocompact Ascoli spaces which are not $k_\mathbb{R}$-spaces and show that each space $X$ can be closely embedded into such a space. Using a different method we prove Hušek's theorem: a Tychonoff space $Y$ is a locally pseudocompact $k_\mathbb{R}$-space iff $X\times Y$ is a $k_\mathbb{R}$-space for each $k_\mathbb{R}$-space $X$. It is proved that $X$ is an $s_\mathbb{R}$-space iff for every locally compact sequential space $K$, each $s$-continuous function $f:X\times K\to\mathbb{R}$ is continuous. |
| title | Functions on products $X \times Y$ with applications to Ascoli spaces, $k_{\mathbb{R}}$-spaces and $s_{\mathbb{R}}$-spaces |
| topic | General Topology 54A05 54B05 54B15 54D60 |
| url | https://arxiv.org/abs/2507.09670 |