Normed versus topological groups: Dichotomy and duality

Normed groups 89The result below generalizes the category version of the Steinhaus Theorem [St] of1920, first stated explicitly by Piccard [Pic1] in 1939, and restated in [Pet1] in 1950; inthe current form it may be regarded as a ‘localized-refinement’ of [RR-TG]. We need adefinition which extends sequential convergence to continuous convergence.Definition (cf. [Mon2]). Let {ψ u : u ∈ I} for I an open interval in R be a family ofhomeomorphisms in H(X). Let u 0 ∈ I. Say that ψ u converges to the identity as u → u 0iflim u→u0 ‖ψ u ‖ = 0.The setting of the next theorem is quite general: homogeneity (relative to H(X)), i.e.all we require is that any point may be transformed to another by a bounded homeomorphismof (X, d).Theorem 6.14 (Generalized Piccard-Pettis Theorem: [Pic1], [Pic2], [Pet1], [Pet2], [BGT,Th. 1.1.1], [BOst-StOstr], [RR-TG], cf. [Kel, Ch. 6 Prb. P]). Let X be a homogenousspace. Suppose that the homeomorphisms ψ u converge to the identity as u → u 0 , and thatA is Baire and non-meagre. Then, for some δ > 0, we haveor, equivalently, for some δ > 0A ∩ ψ u (A) ≠ ∅, for all u with d(u, u 0 ) < δ,A ∩ ψ −1u (A) ≠ ∅, for all u with d(u, u 0 ) < δ.Proof. We may suppose that A = V \M with M meagre and V open. Hence, for anyv ∈ V \M, there is some ε > 0 withB ε (v) ⊆ U.As ψ u → id, there is δ > 0 such that, for u with d(u, u 0 ) < δ, we haveˆd(ψ u , id) < ε/2.SoandHence, for any such u and any y in B ε/2 (v), we haveFor fixed u with d(u, u 0 ) < δ, the setd(ψ u (y), y) < ε/2.W := ψ u (B ε/2 (z 0 )) ∩ B ε/2 (z 0 ) ≠ ∅,W ′ := ψ −1u (B ε/2 (z 0 )) ∩ B ε/2 (z 0 ) ≠ ∅.M ′ := M ∪ ψ u (M) ∪ ψu−1 (M)is meagre. Let w ∈ W \M ′ (or w ∈ W ′ \M ′ , as the case may be). Since w ∈ B ε (z 0 )\M ⊆V \M, we havew ∈ V \M ⊆ A.Similarly, w ∈ ψ u (B ε (z 0 ))\ψ u (M) ⊆ ψ u (V )\ψ u (M). Henceψ −1u (w) ∈ V \M ⊆ A.