拓扑群幂中的可数紧性与紧性的拉姆齐理论变体
Countable compactness in powers of topological groups and Ramsey theoretic variations of compactness
浏览论文内容
中文总结 AI 辅助
该研究针对拓扑群引入级联可数紧性，证明其与有限乘积、可数紧性的关联，构造了多个满足特定级联紧性但不满足可数紧性的拓扑空间与布尔群。
中文摘要 AI 辅助
可数紧性不一定被有限乘积保持。受由ω的有限子集索引的映射的紧性概念启发，我们针对任意障碍（barrier）引入了级联可数紧性的概念。对于$\boldsymbol{\textit{B}=[\boldsymbol{\textit{ω}}]^2}$，这是先前研究过的双可数紧性概念。我们证明，在ZFC公理系统中，若G是豪斯多夫拓扑群且$\boldsymbol{1≤k<ω}$，则G的k-级联可数紧性蕴含G的k次幂是可数紧的。施赖尔障碍（Schreier barrier）的级联可数紧性蕴含G的ω次幂是可数紧的。与之相对，我们构造了一个对每个$\boldsymbol{n<ω}$都为n-级联可数紧的吉洪诺夫（Tychonoff）空间，但其平方不是可数紧的；还构造了一个对每个障碍B都为B-可数紧的豪斯多夫布尔群H，其平方不是可数紧的。我们还构造了一个无平凡收敛序列的豪斯多夫布尔群，它对每个障碍B都为B-级联可数紧。最后，我们得到了βω的子空间X，使得对每个$\boldsymbol{κ<\boldsymbol{\textit{h}}}$和每个$\boldsymbol{n<ω}$，X的κ次幂都是n-级联可数紧的，而X的幂集（exp X）不是伪紧的。
英文摘要
Countable compactness need not be preserved by finite products. Motivated by compactness notions for maps indexed by finite subsets of $ω$, we introduce cascade countable compactness for arbitrary barriers. For $\mathcal B=[ω]^2$, this is the previously studied notion of being doubly countably compact. We show that, in ZFC, if $G$ is a Hausdorff topological group and $1\leq k<ω$, then $k$-cascade countable compactness of $G$ implies that $G^k$ is countably compact. Cascade countable compactness for the Schreier barrier implies that $G^ω$ is countably compact. In contrast, we construct a Tychonoff space that is $n$-cascade countably compact for every $n<ω$ but has a non-countably compact square, and a Hausdorff Boolean group $H$ that is $\mathcal B$-countably compact for every barrier $\mathcal B$ but whose square is not countably compact. We also construct a Hausdorff Boolean group without nontrivial convergent sequences that is $\mathcal B$-cascade countably compact for every barrier $\mathcal B$. Finally, we obtain a subspace $X\subseteqβω$ such that $X^κ$ is $n$-cascade countably compact for every $κ<\mathfrak h$ and every $n<ω$, whereas $\exp X$ is not pseudocompact.
