$κ$-spaces
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912484185604096 |
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| author | Gabriyelyan, Saak Reznichenko, Evgenii |
| author_facet | Gabriyelyan, Saak Reznichenko, Evgenii |
| contents | We say that a Tychonoff space $X$ is a $κ$-space if it is homeomorphic to a closed subspace of $C_p(Y)$ for some locally compact space $Y$. The class of $κ$-spaces is strictly between the class of Dieudonné complete spaces and the class of $μ$-spaces. We show that the class of $κ$-spaces has nice stability properties, that allows us to define the $κ$-completion $κX$ of $X$ as the smallest $κ$-space in the Stone--Čech compactification $βX$ of $X$ containing $X$. For a point $z\inβX$, we show that (1) if $z\in\upsilon X$, then the Dirac measure $δ_z$ at $z$ is bounded on each compact subset of $C_p(X)$, (2) $z\in κX$ iff $δ_z$ is continuous on each compact subset of $C_p(X)$ iff $δ_z$ is continuous on each compact subset of $C_p^b(X)$, (3) $z\in\upsilon X$ iff $δ_z$ is bounded on each compact subset of $C_p^b(X)$. It is proved that $κX$ is the largest subspace $Y$ of $βX$ containing $X$ for which $C_p(Y)$ and $C_p(X)$ have the same compact subsets, this result essentially generalizes a known result of R.~Haydon. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11220 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $κ$-spaces Gabriyelyan, Saak Reznichenko, Evgenii General Topology 54A05, 54B05, 54C35, 54D50 We say that a Tychonoff space $X$ is a $κ$-space if it is homeomorphic to a closed subspace of $C_p(Y)$ for some locally compact space $Y$. The class of $κ$-spaces is strictly between the class of Dieudonné complete spaces and the class of $μ$-spaces. We show that the class of $κ$-spaces has nice stability properties, that allows us to define the $κ$-completion $κX$ of $X$ as the smallest $κ$-space in the Stone--Čech compactification $βX$ of $X$ containing $X$. For a point $z\inβX$, we show that (1) if $z\in\upsilon X$, then the Dirac measure $δ_z$ at $z$ is bounded on each compact subset of $C_p(X)$, (2) $z\in κX$ iff $δ_z$ is continuous on each compact subset of $C_p(X)$ iff $δ_z$ is continuous on each compact subset of $C_p^b(X)$, (3) $z\in\upsilon X$ iff $δ_z$ is bounded on each compact subset of $C_p^b(X)$. It is proved that $κX$ is the largest subspace $Y$ of $βX$ containing $X$ for which $C_p(Y)$ and $C_p(X)$ have the same compact subsets, this result essentially generalizes a known result of R.~Haydon. |
| title | $κ$-spaces |
| topic | General Topology 54A05, 54B05, 54C35, 54D50 |
| url | https://arxiv.org/abs/2507.11220 |