Super-sequences in the arc component of a compact connected group

Abstract

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).