Quasi‐convexly dense and suitable sets in the arc component of a compact group

Abstract

AbstractLet G be an abelian topological group. The symbol \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\widehat{G}$\end{document} denotes the group of all continuous characters \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\chi :G\rightarrow {\mathbb T}$\end{document} 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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\chi \in \widehat{G}$\end{document}, where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\varphi : {\mathbb R}\rightarrow {\mathbb T}={\mathbb R}/{\mathbb Z}$\end{document} 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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$r:\widehat{G}\rightarrow \widehat{G}_a$\end{document} defined by \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$r(\chi )=\chi \upharpoonright _{G_a}$\end{document} for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\chi \in \widehat{G}$\end{document}, 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.