Reflection principle characterizing groups in which unconditionally closed sets are algebraic
Reflection principle characterizing groups in which unconditionally closed sets are algebraic
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DOI：
10.1515/jgt.2008.025
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发表时间：
2008-01-01
影响因子：
0.5
通讯作者：
Shakhmatov, Dmitri
中科院分区：
文献类型：
--
作者：
Dikranjan, Dikran;Shakhmatov, Dmitri
利用反射原理，给出了群G的每个无条件闭子集是代数的充要条件。作为推论，我们证明了当G是Abel群与一族可数群的直积(有时也称为直和)的直积的直积时，总是这样。这是迄今为止已知的最广泛的一类群体，在这些群体中，63年来的马尔可夫问题的答案被证明是肯定的。我们还证明了G的每个无条件闭子集是否是代数的完全由G的可数子群决定，并强调了与不可拓扑化群的本质联系。
We give a necessary and sufficient condition, in terms of a certain :reflection principle, for every unconditionally closed subset of a group G to be algebraic. As a corollary, we prove that this is always the case when G is a direct product of an Abelian group with a direct product (sometimes also called a direct sum) of a family of countable groups. This is the widest class of groups known to date where the answer to the 63-year-old problem of Markov turns out to be positive. We also prove that whether every unconditionally closed subset of G is algebraic or not is completely determined by countable subgroups of G. Essential connections with non-topologizable groups are highlighted.