Normed versus topological groups: Dichotomy and duality

Normed groups 35Definition. A group G ⊂ H(X) acts transitively on a space X if for each x, y in Xthere is g in X such that g(x) = y.The group acts micro-transitively on X if for U a neighbourhood of e in G and x ∈ Xthe set {h(x) : h ∈ U} is a neighbourhood of x.Theorem 3.14 (Effros’ Open Mapping Principle, [Eff]). Let G be a Polish topologicalgroup acting transitively on a separable metrizable space X. The following are equivalent.(i) G acts micro-transitively on X,(ii) X is Polish,(iii) X is of second category.Remark. van Mill [vM1] gives the stronger result for G an analytic group (see Section11 for definition) that (iii) implies (i). See also Section 10 for definitions, references andthe related classical Open Mapping Theorem (which follows from Th. 3.14: see [vM1]).Indeed, van Mill ([vM1]) notes that he uses (i) separately continuous action (see the finalpage of his proof), (ii) the existence of a sequence of symmetric neighbourhoods U n ofthe identity with U n+1 ⊆ U 2 n+1 ⊆ U n , and (iii) U 1 = G (see the first page of his proof).By Th. 2.19 ’ (Birkhoff-Kakutani Normability Theorem) van Mill’s conditions under (ii)specify a normed group, whereas condition (iii) may be arranged by switching to theequivalent norm ||x|| 1 := max{||x||, 1} and then taking U n := {x : ||x|| 1 < 2 −n }. Thus infact one hasTheorem 3.14 ′ (Analytic Effros Open Mapping Principle). For T an analytic normedgroup acting transitively and separately continuously on a separable metrizable space X:if X is non-meagre, then T acts micro-transitively on X.The normed-group result is of interest, as some naturally occurring normed groupsare not complete (see Charatonik et Maćkowiak [ChMa] for Borel normed groups thatare not complete, and [FaSol] for a study of Borel subgroups of Polish groups).Theorem 3.15 (Crimping Theorem). Let T be a Polish space with a complete metric d.Suppose that a closed subgroup G of H u (T ) acts on T transitively, i.e. for any s, t in Tthere is h in G such that h(t) = s. Then for each ε > 0 and t ∈ T, there is δ > 0 suchthat for any s with d T (s, t) < δ, there exists h in G with ‖h‖ H < ε such that h(t) = s.Consequently:(i) if y, z are in B δ (t), then there exists h in G with ‖h‖ H < 2ε such that h(y) = z;(ii) Moreover, for each z n → t there are h n in G converging to the identity such thath n (t) = z n .Proof. As T is Polish, G is Polish, and so by Effros’ Theorem, G acts micro-transitivelyon T ; that is, for each t in T and each ε > 0 the set {h(t) : h ∈ H u (T ) and ‖h‖ H < ε}