Normed versus topological groups: Dichotomy and duality

64 N. H. Bingham and A. J. Ostaszewski(i) W ∩ Z Γ is dense in W ;(ii) ω(t) = 0 for all t ∈ W.In particular, if W = X, then X is topological.For a general W, as above,(i) if e X ∈ W and X is connected and Baire (under its norm topology), then X is atopological group,(ii) if X is separable, connected and topologically complete (in its norm topology), thenX is a topological group.Proof. We first verify that (a)=⇒(b)=⇒(c)=⇒(a).Assume (a). Let ε > 0. Consider a nonempty U ⊆ W. Pick t ∈ U and suppose thatB η (t) ⊆ U with η < ε. By assumption, there is y η = y η (t) such that for some δ = δ(η) < ηwe haved R (tz, zy η ) ≤ η/2 for all ‖z‖ ≤ δ.Then in particular d R (t, y η ) ≤ η/2 < η, and also for all ‖z‖ ≤ δd R (y η z, zy η ) ≤ d R (y η z, tz) + d R (tz, zy η ) = d R (y η , t) + d R (tz, zy η ) ≤ η < ε.Thus y η ∈ B η (t) ∩ C(ε) ⊆ U ∩ C(ε). That is, (b) holds.Assume (b). Consider a non-empty U ⊆ W. Pick t ∈ U and suppose that B η (t) ⊆ U withη < ε/3. By assumption, there is y η = y η (t) ∈ B η (t) such that for some δ = δ(η) < η wehaved R (y η z, zy η ) ≤ η for all ‖z‖ ≤ δ.We prove that y η is a 3η-shifting point and so a ε-shifting point, i.e. thatIndeed, we haved R (y η z, y η ) ≤ 3η for all ‖z‖ ≤ δ.d R (y η z, y η ) ≤ d R (y η z, zy η ) + d R (zy η , y η ) = d R (y η z, zy η ) + d R (z, e)≤ 2η + δ < 3η < ε.Thus, y η ∈ B η (t) ∩ Ω(ε) ⊆ U ∩ Ω(ε). That is, (c) holds.Now suppose that (c) holds. Consider t ∈ W and ε > 0. Suppose that B η (t) ⊆ W withη < ε/2. By assumption, there is y η = y η (t) such that for some δ = δ(η) < η we haveHenced R (y η z, zy η ) ≤ η for all ‖z‖ ≤ δ.d R (tz, zy η ) ≤ d R (tz, y η z) + d R (y η z, zy η ) ≤ 2η < ε.So for y = y η , we have d R (tz, zy) < ε for all ‖z‖ ≤ δ. Thus (a) holds.Now that we have verified the equivalences, suppose that (a) holds.From (c), for t ∈ W and any ε > 0, we have ω(t) ≤ ω δ(ε) (t) ≤ ε. As ε > 0 was arbitrary,we have ω(t) = 0. Hence if W = X, then X = Z Γ and the group is topological. The othertwo conclusions follow from Th. 3.44 and 3.4˙5.