Super-sequences in the Arc Component of a Compact Connected Group

Let G be an abelian topological group. The symbol Ĝ 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) ⊆ φ([−1/4, 1/4]) holds only for the trivial character χ ∈ Ĝ, 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 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 : Ĝ → Ĝa defined by r(χ) = χ ↾Ga for χ ∈ Ĝ, is a topological isomorphism. We also show that an infinite compact group G is connected if and only if its arc component Ga contains a super-sequence S converging to the identity e that generates a dense subgroup of G (equivalently, S \ {e} is an infinite suitable set for G in the sense of Hofmann and Morris).