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D-independent topological groups
Topology and its Applications ( IF 0.6 ) Pub Date : 2021-06-25 , DOI: 10.1016/j.topol.2021.107761
Edgar Márquez Rodríguez , Mikhail Tkachenko

A topological group G with |G|>1 is called d-independent if for every subgroup S of G with |S|<2ω, one can find a countable dense subgroup H of G such that SH={e}. Therefore, d-independent groups are separable and have cardinality at least 2ω. Our main result is a purely algebraic characterization of d-independence in the class of compact metrizable abelian groups. We prove that a compact metrizable abelian group G with |G|>1 is d-independent if and only if for every integer m1, either |mG|=2ω or |mG|=1. This characterization implies that a compact metrizable abelian group is d-independent if and only if it is maximally fragmentable [Comfort and Dikranjan (2014) [4]] iff G an M-group as defined by Dikranjan and Shakhmatov (2016) in [7].

Also we present a characterization of separable metrizable d-independent abelian groups and show that products of separable topological groups can often be d-independent, even if the factors fail to be d-independent.



中文翻译:

D-独立拓扑群

甲拓扑群G ^|G|>1被称为d无关,若对所有子群小号ģ|S。|<2ω,可以发现一个可数稠密子群ħģ使得S。H={电子}. 因此,独立团是可分离的,并至少有基数2ω. 我们的主要结果是对紧凑可度量阿贝尔群类中d独立性的纯代数表征。我们证明了一个紧度量阿贝尔群G ^|G|>1d无关当且仅当对于每个整数1, 任何一个 |G|=2ω 或者 |G|=1. 这种表征意味着一个紧凑的可计量阿贝尔群是d独立的,当且仅当它是最大可碎片化的[Comfort and Dikranjan (2014) [4]] iff G是由 Dikranjan 和 Shakhmatov (2016) 在 [7 ]中定义的M 群]。

此外,我们还介绍了可分离的可计量d独立阿贝尔群的表征,并表明可分离拓扑群的乘积通常可以是d独立的,即使这些因子不能是d独立的。

更新日期:2021-07-02
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