Normed versus topological groups: Dichotomy and duality

66 N. H. Bingham and A. J. OstaszewskiDefinition. A map f : X → Y between metric spaces is quasi-continuous at x if forε > 0 there are a ∈ B X ε (x) and δ > 0 such thatf(u) ∈ B Y ε (f(x)), for all u ∈ B X δ (a).The following result connects quasi-continuity of left-shifts λ t with ε-shifting.Theorem 3.52. Let X be a normed group X.(i) The left-shift λ t (x), as a self-map of X under the right norm topology, is quasicontinuousat any point/all points x iff for every ε > 0 there are y = y(ε) and δ = δ(ε) > 0such thatd L (t, y(ε)) < ε, and d R (tz, y(ε)) < ε, for ‖z‖ < δ.(ii) If λ t is quasi-continuous, then t is an ε-shifting point for each ε > 0.(iii) In these circumstance, γ t has zero oscillation, hence γ t and so λ t is continuous.Proof. (i) This is a routine transcription of the last definition, so we omit the details.The point y of the Theorem is obtained from the point a of the definition via y := ta −1 .(ii) This conclusion come from taking z = e and applying the triangle inequality to obtaind R (tz, t) < 2ε, for ‖z‖ < δ(ε).(iii) It follows from (ii) that ω(t) = 0, so that γ t is continuous at e and hence everywhere;λ t is then continuous, being a composition of continuous functions, since tx = ρ t (γ t (x)).Of course in the setting above t m := y(1/m) converges to t under both norm topologies.This gives a restatement of a preceding result (Th. 3.46).Theorem 3.53. In a normed group X with right norm topology, if for a dense set of tthe left-shifts λ t (x) are quasi-continuous, then the normed group is topological.Alternatively, note that under the current assumptions the topological centre Z Γ isdense, and being closed is the whole of X. Our closing comment addresses the openingissue of this subsection – converging subsequences – in terms of subcontinuity. We recalla result of Bouziad, again specialized to our metric context.Theorem 3.54 ([Bou2, Lemma 2.4]). For f : X → Y a quasi-continuous map betweenmetric spaces with X Baire, the set of subcontinuity points of f is a dense subset of X.