Uniform homeomorphisms between $C_p^*$-spaces preserve pseudocompactness
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| Format: | Preprint |
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2026
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| author | Valov, Vesko |
| author_facet | Valov, Vesko |
| contents | For any Tychonoff space $X$ let $C_p(X)$ (resp., $C^*_p(X)$) be the set of all continuous (resp., and bounded) functions on $X$ with the pointwise convergence topology. Given Tychonoff spaces $X$ and $Y$, Uspenskij \cite{us} proved that if $C_p(X)$ is uniformly homeomorphic to $C_p(Y)$, then $X$ is pseudocompact if and only if $Y$ is pseudocompact. The author and Vuma \cite{valvu} have shown that linear homeomorphisms between $C_p^*(X)$ and $C_p^*(Y)$ preserve pseudocompactness. Recently Baars-van Mill-Tkachuk \cite{bmt} gave another proof of that result and raised the question if the same remains true provided $C_p^*(X)$ and $C_p^*(Y)$ are uniformly homeomorphic. In the present paper we answer that question positively. This, together with a result of Krupski \cite{k}, implies that $κ$-pseudocompactness is also preserved by uniform homeomorphisms between $C_p^*$-spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_25827 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniform homeomorphisms between $C_p^*$-spaces preserve pseudocompactness Valov, Vesko General Topology Primary 54C35, Secondary 54D30 For any Tychonoff space $X$ let $C_p(X)$ (resp., $C^*_p(X)$) be the set of all continuous (resp., and bounded) functions on $X$ with the pointwise convergence topology. Given Tychonoff spaces $X$ and $Y$, Uspenskij \cite{us} proved that if $C_p(X)$ is uniformly homeomorphic to $C_p(Y)$, then $X$ is pseudocompact if and only if $Y$ is pseudocompact. The author and Vuma \cite{valvu} have shown that linear homeomorphisms between $C_p^*(X)$ and $C_p^*(Y)$ preserve pseudocompactness. Recently Baars-van Mill-Tkachuk \cite{bmt} gave another proof of that result and raised the question if the same remains true provided $C_p^*(X)$ and $C_p^*(Y)$ are uniformly homeomorphic. In the present paper we answer that question positively. This, together with a result of Krupski \cite{k}, implies that $κ$-pseudocompactness is also preserved by uniform homeomorphisms between $C_p^*$-spaces. |
| title | Uniform homeomorphisms between $C_p^*$-spaces preserve pseudocompactness |
| topic | General Topology Primary 54C35, Secondary 54D30 |
| url | https://arxiv.org/abs/2604.25827 |