Normed versus topological groups: Dichotomy and duality

Normed groups 81Theorem 6.1 (Category Embedding Theorem, CET). Let X be a topological space. Supposegiven homeomorphisms ψ n : X → X for which the weak category convergence condition(wcc) is met. Then, for any non-meagre Baire set T, for quasi all t ∈ T, there isan infinite set M t such that{ψ m (t) : m ∈ M t } ⊆ T.Proof. Take T Baire and non-meagre. We may assume that T = U\M with U nonemptyand open and M meagre. Let V ⊆ U satisfy (wcc). Since the functions h n arehomeomorphisms, the setM ′ := M ∪ ⋃ n h−1 n (M)is meagre. Writing ‘i.o.’ for ‘infinitely often’, putW = h(V ) := ⋂ ⋃V ∩k∈ω n≥k h−1 n (V ) = lim sup[h −1n (V ) ∩ V ]= {x : x ∈ h −1n (V ) ∩ V i.o.} ⊆ V ⊆ U.So for t ∈ W we have t ∈ V andv m := h m (t) ∈ V, (*)for infinitely many m – for m ∈ M t , say. Now W is co-meagre in V. IndeedV \W = ⋃ ⋂Vk∈ω n≥k \h−1 n (V ),which by (wcc) is meagre.Take t ∈ W \M ′ ⊆ U\M = T, as V ⊆ U and M ⊆ M ′ . Thus t ∈ T. For m ∈ M t , wehave t /∈ h −1m (M), since t /∈ M ′ and h −1m (M) ⊆ M ′ ; but v m = h m (t) so v m /∈ M. By (*),v m ∈ V \M ⊆ U\M = T. Thus {h m (t) : m ∈ M t } ⊆ T for t in a co-meagre set.To deduce that quasi-all t ∈ T satisfy the conclusion of the theorem, put S := T \h(T );then S is Baire and S ∩ h(T ) = ∅. If S is non-meagre, then by the preceeding argumentthere are s ∈ S and an infinite M s such that {h m (s) : m ∈ M s } ⊆ S, i.e. s ∈ h(S) ⊆ h(T ),a contradiction. (This last step is an implicit appeal to a generic dichotomy – see Th.5.4.)Examples. In R we may consider ψ n (t) = t + z n with z n → z 0 := 0. It is shownin [BOst-LBII] that for this sequence the condition (wcc) is satisfied in both the usualtopology and the density topology on R. This remains true in R d , where the specificinstance of the theorem is referred to as the Kestelman-Borwein-Ditor Theorem; see thenext section ([Kes], [BoDi]; compare also the Oxtoby-Hoffmann-Jørgensen zero-one lawfor Baire groups, [HJ, p. 356], [Oxt1, p. 85], cf. [RR-01]). In fact in any metrizable group Xwith right-invariant metric d X , for a null sequence tending to the identity z n → z 0 := e X ,the mapping defined by ψ n (x) = z n x converges to the identity (see [BOst-TRI], Corollaryto Ford’s Theorem); here too (wcc) holds. This follows from the next result, which extendsthe proof of [BOst-LBII]; cf. Theorem 7.5.