Compact subspaces of the space of separately continuous functions with the cross-uniform topology
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913633379811328 |
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| author | Maslyuchenko, Oleksandr Myronyk, Vadym Ivasiuk, Roman |
| author_facet | Maslyuchenko, Oleksandr Myronyk, Vadym Ivasiuk, Roman |
| contents | We consider two natural topologies on the space $S(X\times Y,Z)$ of all separately continuous functions defined on the product of two topological spaces $X$ and $Y$ and ranged into a topological or metric space $X$. These topologies are the cross-open topology and the cross-uniform topology. We show that these topologies coincides if $X$ and $Y$ are pseudocompacts and $Z$ is a metric space. We prove that a compact space $K$ embeds into $S(X\times Y,Z)$ for infinite compacts $X$, $Y$ and a metrizable space $Z\supseteq\mathbb{R}$ if and only if the weight of $K$ is less than the sharp cellularity of both spaces $X$ and $Y$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05705 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Compact subspaces of the space of separately continuous functions with the cross-uniform topology Maslyuchenko, Oleksandr Myronyk, Vadym Ivasiuk, Roman General Topology 54C35 (Primary) 54D30, 54A25, 54C08, 54C25 (Secondary) We consider two natural topologies on the space $S(X\times Y,Z)$ of all separately continuous functions defined on the product of two topological spaces $X$ and $Y$ and ranged into a topological or metric space $X$. These topologies are the cross-open topology and the cross-uniform topology. We show that these topologies coincides if $X$ and $Y$ are pseudocompacts and $Z$ is a metric space. We prove that a compact space $K$ embeds into $S(X\times Y,Z)$ for infinite compacts $X$, $Y$ and a metrizable space $Z\supseteq\mathbb{R}$ if and only if the weight of $K$ is less than the sharp cellularity of both spaces $X$ and $Y$. |
| title | Compact subspaces of the space of separately continuous functions with the cross-uniform topology |
| topic | General Topology 54C35 (Primary) 54D30, 54A25, 54C08, 54C25 (Secondary) |
| url | https://arxiv.org/abs/2406.05705 |