Bounded uniform homeomorphisms between Cp*-spaces preserve pseudocompactness
Abstract
For any Tychonoff space X let Cp(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 Cp(X) is uniformly homeomorphic to Cp(Y), then X is pseudocompact if and only if Y is pseudocompact. The second author and Vuma \cite{valvu} have shown that linear homeomorphisms between Cp*(X) and Cp*(Y) also 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 Cp*(X) and Cp*(Y) are uniformly homeomorphic. In the present paper we introduce the notion of bounded uniformly continuous maps and show that every bounded uniform homeomorphism between Cp*(X) and Cp*(Y) preserve pseudocompactness. Moreover, we show that if there is a uniform homeomorphism φ:Cp*(X)→ Cp*(Y), where X is pseudocompact and Y is normal, then Y is σ-pseudocompact, i.e. Y is a countable union of closed pseudocompact subsets. It is also shown that a continuous linear map between Cp-spaces is norm-bounded if and only if it is bounded in our sense.
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BibTeX
@misc{krupski2026bounded,
title = {{Bounded uniform homeomorphisms between $C_p^*$-spaces preserve pseudocompactness}},
author = {Mikolaj Krupski and Vesko Valov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.23034},
eprint = {2607.23034},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.23034},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bounded uniform homeomorphisms between C_p^*-spaces preserve pseudocompactness
AU - Mikolaj Krupski
AU - Vesko Valov
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2607.23034
UR - https://arxiv.org/abs/2607.23034
AB - 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 second author and Vuma \cite{valvu} have shown that linear homeomorphisms between C_p^*(X) and C_p^*(Y) also 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 introduce the notion of bounded uniformly continuous maps and show that every bounded uniform homeomorphism between C_p^*(X) and C_p^*(Y) preserve pseudocompactness. Moreover, we show that if there is a uniform homeomorphism φ:C_p^*(X)→ C_p^*(Y), where X is pseudocompact and Y is normal, then Y is σ-pseudocompact, i.e. Y is a countable union of closed pseudocompact subsets. It is also shown that a continuous linear map between C_p-spaces is norm-bounded if and only if it is bounded in our sense.
ER -