Quasi-covexly dense and suitable sets in the arc component of a compact group
Quasi-covexly dense and suitable sets in the arc component of a compact group
复制标题
紧群弧分量中的拟凸稠密适集
DOI：
10.1002/mana.201010013
复制
发表时间：
2012
影响因子：
1
通讯作者：
D.Shakhmatov
中科院分区：
文献类型：
--
作者：
D.Dikranjan;D.Shakhmatov
设G是交换拓扑群.符号\hat{G}表示具有紧开拓扑的所有连续特征标\chi：G --> T的群。G的子集E在G中称为qc-稠密的，只要\chi（E）\subseteq \phi（[-1/4，1/4]）仅对平凡特征\chi \in \hat{G}成立，其中\phi：R --> T = R/Z是标准同态。超序列是一个非空的紧致豪斯多夫空间S，它至多有一个非孤立点（S收敛到这个点）。证明了无限紧阿贝尔群G是连通的当且仅当其弧分支G_a包含收敛到0的超序列且在G中是qc-稠密的。这给出了Aussenhofer最近的一个定理：对于连通的局部紧交换群G，由r（\chi）= \chi\restriction_{G_a}定义的限制同态r：\hat {G} --> \hat{G}_a对于\chi \in \hat{G}是拓扑同构。我们还证明了无限紧群G是连通的当且仅当它的弧分量G_a包含收敛到单位元e的超序列S，该超序列S生成G的稠密子群（等价地，S \setminus {e}是一个无限合适的集合）在霍夫曼和莫里斯的意义上，G）。
Let G be an abelian topological group. The symbol \hat{G} denotes the group of all continuous characters \chi : G --> T endowed with the compact open topology. A subset E of G is said to be qc-dense in G provided that \chi(E) \subseteq \phi([-1/4,1/4]) holds only for the trivial character \chi \in \hat{G}, where \phi : R --> T = R/Z is the canonical homomorphism. A super-sequence is a non-empty compact Hausdorff space S with at most one non-isolated point (to which S converges). We prove that an infinite compact abelian group G is connected if and only if its arc component G_a contains a super-sequence converging to 0 that is qc-dense in G. This gives as a corollary a recent theorem of Aussenhofer: For a connected locally compact abelian group G, the restriction homomorphism r : \hat{G} --> \hat{G}_a defined by r(\chi) = \chi\restriction_{G_a} for \chi \in \hat{G}, is a topological isomorphism. We also show that an infinite compact group G is connected if and only if its arc component G_a contains a super-sequence S converging to the identity e that generates a dense subgroup of G (equivalently, S \setminus {e} is an infinite suitable set for G in the sense of Hofmann and Morris).