Normed versus topological groups: Dichotomy and duality

Normed groups 74 (Topological permutation). For π ∈ Auth(X), i.e. a homeomorphism and x fixed,note that for any ε > 0 there is δ = δ(ε) > 0 such thatprovided d(x, y) < δ, i.e.d π (x, y) = d(π(x), π(y)) < ε,B δ (x) ⊂ B π ε (x).Take ξ = π(x) and write η = π(y); there is µ > 0 such thatd(x, y) = d π −1(ξ, η) = d(π −1 (ξ), π −1 (η)) < ε,provided d π (x, y) = d(π(x), π(y)) = d(ξ, η) < µ, i.e.B π µ(x) ⊂ B ε (x).Thus the topology generated by d π is the same as that generated by d. This observationapplies to all the previous examples provided the permutations are homeomorphisms (e.g.if X is a topological group under d X ). Note that for d X right-invariant‖x‖ π = ‖π(x)π(e) −1 ‖.5. For g ∈ Auth(X), h ∈ X, the bijection π(x) = g(ρ h (x)) = g(xh) is a homeomorphismprovided right-shifts are continuous. We refer to this as the shifted g-h-permutation metricwhich has the associated g-h-shifted normd X g-h(x, y) = d X (g(xh), g(yh)),‖x‖ g-h = d X (g(xh), g(h)).6 (Equivalent Bounded norm). Set d b (x, y) = min{d X (x, y), 1}. Then d b is an equivalentmetric (cf. [Eng, Th. 4.1.3, p. 250]). We refer toas the equivalent bounded norm.‖x‖ b := d b (x, e) = min{d X (x, e), 1} = min{‖x‖, 1}7. For A = Auth(X) the evaluation pseudo-metric at x on A is given byand sois a pseudo-norm.d A x (f, g) = d X (f(x), g(x)),‖f‖ x = d A x (f, id) = d X (f(x), x)Definition (Refinements). 1 (cf. [GJ, Ch. 15.3] which works with pseudometrics). Let∆ = {d X i : i ∈ I} be a family of metrics on a group X. The weak (Tychonov) ∆-refinementtopology on X is defined by reference to the local base at x obtained by finite intersectionsof ε-balls about x :⋂i∈F Bi ε(x), for F finite, i.e. Bε i1(x) ∩ ... ∩ Bε in(x), if F = {i 1 , ..., i n },