Quasi‐convexly dense and suitable sets in the arc component of a compact group |
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Authors: | Dikran Dikranjan Dmitri Shakhmatov |
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Institution: | 1. Dipartimento di Matematica e Informatica, Università di Udine, via delle Scienze, 206 ‐ 33100 Udine, Italy;2. Division of Mathematics, Physics and Earth Sciences, Graduate School of Science and Engineering, Ehime University, Matsuyama 790‐8577, Japan |
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Abstract: | Let G be an abelian topological group. The symbol $\widehat{G}Let G be an abelian topological group. The symbol $\widehat{G}$ denotes the group of all continuous characters $\chi :G\rightarrow {\mathbb T}$ endowed with the compact open topology. A subset E of G is said to be qc‐dense in G provided that χ(E)?φ(? 1/4, 1/4]) holds only for the trivial character $\chi \in \widehat{G}$, where $\varphi : {\mathbb R}\rightarrow {\mathbb T}={\mathbb R}/{\mathbb Z}$ 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 Ga 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 $r:\widehat{G}\rightarrow \widehat{G}_a$ defined by $r(\chi )=\chi \upharpoonright _{G_a}$ for $\chi \in \widehat{G}$, is a topological isomorphism. We show that an infinite compact group G is connected if and only if its arc component Ga contains a super‐sequence converging to the identity that is qc‐dense in G and generates a dense subgroup of G. We also offer a short alternative proof of the result of Hofmann and Morris on the existence of suitable sets of minimal size in the arc component of a compact connected group. |
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Keywords: | Compact group abelian group connected space arc component suitable set dual group MSC (2010) Primary: 22C05 Secondary: 22B05 |
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