紧连通群弧分支中的超序列
Super-sequences in the arc component of a compact connected group
摘要：
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χ：G --> T的群。如果x（E）\subseteq ≠（[-1/4，1/4]）仅对平凡特征标χ \in \hat{G}成立，则称G的子集E在G中是qc-稠密的，其中x：R --> T = R/Z是标准同态.一个超序列是一个非空的紧致T2空间S，其中至多有一个非孤立点（S收敛到该点）。证明了无限紧阿贝尔群G是连通的当且仅当它的弧分支G_a包含一个收敛到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：对于连通的局部紧阿贝尔群G，由r（χ）= χ \restriction_{G_a}定义的限制同态r：\hat{G} --> \hat{G}_a，对于χ \in \hat{G}，是拓扑同构。我们还证明了一个无限紧群G是连通的当且仅当它的弧分支G_a包含一个超序列S，S收敛于生成G的一个稠密子群的恒等式e（等价地，S \setminus {e}是G在Hofmann和Morris意义下的一个无限适集）.
英文摘要：
Let G be an abelian topological group. The symbol \hat{G} denotes the group of all continuous characters χ: G --> T endowed with the compact open topology. A subset E of G is said to be qc-dense in G provided that χ(E) \subseteq ϕ([-1/4,1/4]) holds only for the trivial character χ\in \hat{G}, where ϕ: 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(χ) = χ\restriction_{G_a} for χ\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).
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