Countable spaces, realcompactness, and the pseudointersection number
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917840987095040 |
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| author | Agostini, Claudio Medini, Andrea Zdomskyy, Lyubomyr |
| author_facet | Agostini, Claudio Medini, Andrea Zdomskyy, Lyubomyr |
| contents | All spaces are assumed to be Tychonoff. Given a realcompact space $X$, we denote by $\mathsf{Exp}(X)$ the smallest infinite cardinal $κ$ such that $X$ is homeomorphic to a closed subspace of $\mathbb{R}^κ$. Our main result shows that, given a cardinal $κ$, the following conditions are equivalent: $(1)$ There exists a countable crowded space $X$ such that $\mathsf{Exp}(X)=κ$, $(2)$ $\mathfrak{p}\leqκ\leq\mathfrak{c}$. In fact, in the case $\mathfrak{d}\leqκ\leq\mathfrak{c}$, every countable dense subspace of $2^κ$ provides such an example. This will follow from our analysis of the pseudocharacter of countable subsets of products of first-countable spaces. Finally, we show that a scattered space of weight $κ$ has pseudocharacter at most $κ$ in any compactification. This will allow us to calculate $\mathsf{Exp}(X)$ for an arbitrary (that is, not necessarily crowded) countable space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_17984 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Countable spaces, realcompactness, and the pseudointersection number Agostini, Claudio Medini, Andrea Zdomskyy, Lyubomyr General Topology Logic 54D60, 03E17, 54G12 All spaces are assumed to be Tychonoff. Given a realcompact space $X$, we denote by $\mathsf{Exp}(X)$ the smallest infinite cardinal $κ$ such that $X$ is homeomorphic to a closed subspace of $\mathbb{R}^κ$. Our main result shows that, given a cardinal $κ$, the following conditions are equivalent: $(1)$ There exists a countable crowded space $X$ such that $\mathsf{Exp}(X)=κ$, $(2)$ $\mathfrak{p}\leqκ\leq\mathfrak{c}$. In fact, in the case $\mathfrak{d}\leqκ\leq\mathfrak{c}$, every countable dense subspace of $2^κ$ provides such an example. This will follow from our analysis of the pseudocharacter of countable subsets of products of first-countable spaces. Finally, we show that a scattered space of weight $κ$ has pseudocharacter at most $κ$ in any compactification. This will allow us to calculate $\mathsf{Exp}(X)$ for an arbitrary (that is, not necessarily crowded) countable space. |
| title | Countable spaces, realcompactness, and the pseudointersection number |
| topic | General Topology Logic 54D60, 03E17, 54G12 |
| url | https://arxiv.org/abs/2310.17984 |