发表机构
Universitat Jaume I（豪梅一世大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究阿贝尔拓扑群的对偶性，引入$\omega$-桶状群并证明其蕴含$\aleph_0$-桶状性质，解决开放问题并刻画完全有界群紧子集有限性与对偶群类贝尔性质的关系。
AI 中文摘要
本文通过深入探讨阿贝尔群$G$的代数与拓扑性质及其对偶群之间的相互作用，研究阿贝尔拓扑群的对偶性。特别地，我们推广了$g$-桶状群的概念，引入并研究了$\kappa$-桶状群和$\omega$-桶状群类。我们的第一个主要结果确立了每个$\omega$-桶状群都是$\aleph_0$-桶状群（即，如果对偶群中的每个弱收敛序列是等度连续的，那么对偶群的每个可度量化弱紧子集也是等度连续的）。由此，我们推断出在可度量化紧空间$K$上的自由拓扑阿贝尔群$A(K)$由其任意稠密子集生成的子群决定。此外，我们证明如果$G$是一个$\sigma$-紧的$\omega$-桶状群，则其配备紧开拓扑的对偶群$\widehat{G}_\kappa$是序列完备的（若$G$是半紧的，则是完备的）。我们还通过构造一个显式的$\omega$-桶状可度量化群但不满足$g$-桶状性质的例子，解决了Trigos-Arrieta提出的一个开放问题。最后，我们研究了缺乏无限紧子集的群的对偶化，证明对于完全有界阿贝尔群$G$，$G$的每个紧子集是有限的当且仅当其对偶群$\widehat{G}$（赋予有限开拓扑）是无序类贝尔的。
英文摘要
This article investigates the duality of abelian topological groups by delving into the interplay between the algebraic and topological properties of a group $G$ and those of its dual group. In particular, extending the notion of a $g$-barrelled group, we introduce and study the classes of $κ$-barrelled and $ω$-barrelled groups. Our first main result establishes that every $ω$-barrelled group is $\aleph_0$-barrelled (i.e., if every weakly convergent sequence in the dual group is equicontinuous, then every metrizable weakly compact subset of the dual group is also equicontinuous). From this, we deduce that the free topological abelian group $A(K)$ over a metrizable compact space $K$ is determined by the subgroup generated by any of its dense subsets. Furthermore, we prove that if $G$ is a $σ$-compact, $ω$-barrelled group, then its dual group $\widehat{G}_κ$ equipped with the compact-open topology is sequentially complete (complete if $G$ is hemicompact). We also solve an open question posed by Trigos-Arrieta by constructing an explicit example of an $ω$-barrelled metrizable group that fails to be $g$-barrelled. Finally, we investigate the dualization of groups lacking infinite compact subsets, proving that for a totally bounded abelian group $G$, every compact subset of $G$ is finite if and only if its dual group $\widehat{G}$, endowed with the finite-open topology, is unordered Baire-like.
