紧群的度量化准则
Metrization criteria for compact groups in terms of their dense subgroups
摘要：
根据Comfort，Raczkowski和Trigos-Arrieta，如果一个紧阿贝尔群G的限制同态G^ --> D^是一个拓扑同构，则G的稠密子群D决定G。我们在D上引入了确定G所必需的四个条件，并解决了以下问题：如果对G的每个稠密（或G_δ-稠密）子群D，其中一个条件成立，那么G一定是度量化的吗？特别地，我们证明了（在ZFC中）由所有G_δ-稠密子群所决定的紧阿贝尔群是可度量化的，从而解决了Hernandez，Macario和Trigos-Arrieta提出的一个问题.（在连续统假设CH的附加假设下，最近布鲁格拉、查斯科、多明格斯、特卡琴科和特里戈斯-阿列塔证明了同样的陈述。）作为一种工具，我们给出了一种构造权为ω_1的紧群中不含不可数紧子集的G_δ-稠密子群的方法（在ZFC中）.构造是微妙的，因为在ZFC的某些模型中，这些子群必须具有非平凡的收敛序列。
英文摘要：
According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism G^ --> D^ of the dual groups is a topological isomorphism. We introduce four conditions on D that are necessary for it to determine G and we resolve the following question: If one of these conditions holds for every dense (or G_δ-dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined by all its G_δ-dense subgroups is metrizable, thereby resolving a question by Hernandez, Macario and Trigos-Arrieta. (Under the additional assumption of the Continuum Hypothesis CH, the same statement was proved recently by Bruguera, Chasco, Dominguez, Tkachenko and Trigos-Arrieta.) As a tool, we develop a machinery for building G_δ-dense subgroups without uncountable compact subsets in compact groups of weight ω_1 (in ZFC). The construction is delicate, as these subgroups must have non-trivial convergent sequences in some models of ZFC.
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