Uniform homeomorphisms between $C_p^*$-spaces preserve pseudocompactness

Abstract

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 $\kappa$-pseudocompactness is also preserved by uniform homeomorphisms between $C_p^*$-spaces.