Normed versus topological groups: Dichotomy and duality

Normed groups 57Proof. To apply Th. 11.11 or Th. 11.12 we need to interpret inversion as a homomorphismbetween normed groups. To this end, define X ∗ = (X, ∗, d ∗ ) to be the metric group withunderlying set X with multiplication x∗y := yx and metric d ∗ (x, y) = d R (x, y). Then X ∗is isometric with (X, ·, d R ) under the identity and d ∗ is left-invariant, since d ∗ (x∗y, x∗z) =d R (yx, zx) = d R (y, z) = d ∗ (y, z). Thus X ∗ is separable and topologically complete. Nowf : X → X ∗ defined by f(x) = x −1 is a homomorphism, which is Borel/analytic (bythe isometry). Hence f is continuous and so is right-to-right continuous. Now by theEquivalence Theorem (Th. 3.4), the normed group X is topological.For the connection between continuity, openness and the closed graph property ofhomomorphisms, which we just exploited, see [Pet3] and the discussion in [Pet4, esp.VIII]. For related work see Solecki and Srivastava [SolSri], where the group is Baire,separable, metrizable with continuous right-shifts ρ t (s) = st and has Baire-measurableleft-shifts λ s (t) = st. Before leaving the issue of automatic continuity, we note that Th.3.40 and 3.41 have analogues in locally compact, normed group having the Heine-Borelproperty (i.e. a set is compact iff it is closed and norm-bounded) – see [Ost-LB3]. Afurther automatic result that (X, d R ) is a topological group is derived in [Ost-Joint]from the hypotheses that (X, d S ) is non-meagre and (X, d S ) is Polish. (See also [Ost-AB]for the non-separable case which requires further conditions involving the notion of σ-discreteness.)We now study the oscillation function in a normed group setting.Definition. We putω(t) = lim δ↘0 ω δ (t), where ω δ (t) := sup ‖z‖≤δ ‖γ t (z)‖,and call ω(·) the oscillation function of the group-norm. (We will see in Prop. 3.42 thatthese are finite quantities.) If ω(t) < ε, then ω δ (t) < ε, for some δ > 0. In the light ofthis, we will need to refer to the related setsΩ(ε) : = {t : ω(t) < ε},Ω δ (ε) := {t : ω δ (t) < ε},Λ δ (ε) : = {t : d(t, tz) ≤ ε for all ‖z‖ ≤ δ},so that for d = d R we have Ω δ (ε) ⊆ Λ δ (ε) andΩ(ε) ⊂ ⋃ δ∈R +Λ δ (ε) ⊂ Ω(2ε).(cover)It is convenient on occasion to allow the d in Λ δ (ε) to be a general metric compatiblewith the topology of X (not necessarily right-invariant).Remarks. 1. Of course if ω(t) = 0, then γ t is continuous.2. For fixed z and ε > 0, the setsF ε (z) = {t : d(t, tz) ≤ ε}, and G ε (z) = {t : d(t, tz) < ε},