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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.18027 |
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| _version_ | 1866916350177312768 |
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| author | Ríos-Herrejón, Alejandro Tamariz-Mascarúa, Ángel |
| author_facet | Ríos-Herrejón, Alejandro Tamariz-Mascarúa, Ángel |
| contents | We present new results regarding calibers in the function spaces $C_p(X)$. Our main theorem is that $C_p(X)$ is strongly Šanin whenever $X$ is a submetrizable space; this improves an earlier result due to Tkachuk: $C_p(X)$ is Šanin whenever $X$ is a submetrizable space. Moreover, we give sufficient conditions to characterize the calibers of $C_p(X)$ when $X$ is a topological sum, and we calculate the calibers of $C_p(X)$ when $X = \prod_{ξ< λ}X_ξ$ is a product of non-trivial Tychonoff spaces with $i$-weight $\leq λ$. Furthermore, we calculate the calibers of $C_p(X)$ when $X$ is an interval of ordinals and when $X$ is the one-point $λ$-Lindelöf extension of a discrete space of cardinality $\geq λ$. This allows to give examples of compact Hausdorff spaces $Z$ such that $iw(Z)=κ^{+}$ and $C_p(Z^κ)$ does not have caliber $iw(Z)$; and examples of spaces $\{Z_α: α<cf(κ)\}$ such that $κ$ is a caliber for $C_p(Z_α)$ whenever $α<cf(κ)$ but it is not a caliber for $C_p(\bigoplus_{α<cf(κ)} Z_α)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18027 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On calibers for $C_p(X)$ Ríos-Herrejón, Alejandro Tamariz-Mascarúa, Ángel General Topology We present new results regarding calibers in the function spaces $C_p(X)$. Our main theorem is that $C_p(X)$ is strongly Šanin whenever $X$ is a submetrizable space; this improves an earlier result due to Tkachuk: $C_p(X)$ is Šanin whenever $X$ is a submetrizable space. Moreover, we give sufficient conditions to characterize the calibers of $C_p(X)$ when $X$ is a topological sum, and we calculate the calibers of $C_p(X)$ when $X = \prod_{ξ< λ}X_ξ$ is a product of non-trivial Tychonoff spaces with $i$-weight $\leq λ$. Furthermore, we calculate the calibers of $C_p(X)$ when $X$ is an interval of ordinals and when $X$ is the one-point $λ$-Lindelöf extension of a discrete space of cardinality $\geq λ$. This allows to give examples of compact Hausdorff spaces $Z$ such that $iw(Z)=κ^{+}$ and $C_p(Z^κ)$ does not have caliber $iw(Z)$; and examples of spaces $\{Z_α: α<cf(κ)\}$ such that $κ$ is a caliber for $C_p(Z_α)$ whenever $α<cf(κ)$ but it is not a caliber for $C_p(\bigoplus_{α<cf(κ)} Z_α)$. |
| title | On calibers for $C_p(X)$ |
| topic | General Topology |
| url | https://arxiv.org/abs/2403.18027 |