$C_p^*$-空间之间的有界一致同胚保持伪紧性
Bounded uniform homeomorphisms between $C_p^*$-spaces preserve pseudocompactness
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中文总结 AI 辅助
研究\(C_p^*(X)\)与\(C_p^*(Y)\)一致同胚时是否保持伪紧性这一问题，引入有界一致连续映射概念，证明\(C_p^*(X)\)与\(C_p^*(Y)\)间有界一致同胚保持伪紧性，且明确\(C_p\)-空间间连续线性映射有界性的等价条件。
中文摘要 AI 辅助
对于任意吉洪诺夫空间\(X\)，令\(C_p(X)\)（相应地，\(C_p^*(X)\)）为\(X\)上所有具有点态收敛拓扑的连续（相应地，有界）函数的集合。给定吉洪诺夫空间\(X\)和\(Y\)，乌斯宾斯基证明若\(C_p(X)\)与\(C_p(Y)\)一致同胚，则\(X\)是伪紧的当且仅当\(Y\)是伪紧的。第二作者和武马表明\(C_p^*(X)\)与\(C_p^*(Y)\)之间的线性同胚也保持伪紧性。最近巴尔斯 - 范·米尔 - 特卡丘克给出了该结果的另一种证明并提出问题：若\(C_p^*(X)\)与\(C_p^*(Y)\)一致同胚，该结论是否仍然成立。本文引入有界一致连续映射的概念并表明\(C_p^*(X)\)与\(C_p^*(Y)\)之间的每个有界一致同胚都保持伪紧性。还表明\(C_p\)-空间之间的连续线性映射是范数有界的当且仅当它在我们的意义下是有界的。
英文摘要
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. 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.
