Normed versus topological groups: Dichotomy and duality

118 N. H. Bingham and A. J. OstaszewskiBut X is non-meagre, so for some n the set T a n is non-meagre, and so too is T (as rightshiftsare homeomorphisms). By assumption f is Baire. Thus T is Baire and non-meagre.By the Squared Pettis Theorem (Th. 5.8), (T T −1 ) 2 contains a ball B δ (e X ). Thus we haveB δ (e X ) ⊆ (T T −1 ) 2 ⊆ f −1 [B ε/4 (e Y ) 4 ] = f −1 [B ε (e Y )].Theorem 11.12 (Souslin-graph Theorem, Schwartz [Schw], cf. [Jay-Rog, p.50]). Let Xand Y be normed groups with Y a K-analytic and X non-meagre. If f : X → Y is ahomomorphism with Souslin-F(X × Y ) graph, then f is continuous.Proof. This follows from Theorems 11.9 and 11.11.Corollary 11.13 (Generalized Jones Theorem: Thinned Souslin-graph Theorem). LetX and Y be topological groups with X non-meagre and Y a K-analytic set. Let S bea K-analytic set spanning X and f : X → Y a homomorphism with restriction to Scontinous on S. Then f is continuous.Proof. Since f is continuous on S, the graph {(x, y) ∈ S × Y : y = f(x)} is closed inS × Y and so is K-analytic by [Jay-Rog, Th. 2.5.3]. Now y = f(x) iff, for some n ∈ N,there is (y 1 , ..., y n ) ∈ Y n and (s 1 , ..., s n ) ∈ S n such that x = s 1 · ... · s n , y = y 1 · ... · y n ,and, for i = 1, .., n, y i = f(s i ). Thus G := {(x, y) : y = f(x)} is K-analytic. Formally,whereandG = pr X×Y[⋃n∈N[M n ∩ (X × Y × S n × Y n ) ∩ ⋂ i≤n G i,n]],M n := {(x, y, s 1 , ...., s n , y 1 , ..., y n ) : y = y 1 · ... · y n and x = s 1 · ... · s n },G i,n := {(x, y, s 1 , ...., s n , y 1 , ..., y n ) ∈ X × Y × X n × Y n : y i = f(s i )}, for i = 1, ..., n.Here each set M n is closed and each G i,n is K-analytic. Hence, by the Intersection andProjection Theorems (Th. 11.3 and 11.4), the graph G is K-analytic. By the Souslingraphtheorem f is thus continuous.This is a new proof of the Jones Theorem. We now consider results for the morespecial normed group context. Here again one should note the corollary of [HJ, Th. 2.3.6p. 355] that a normed group which is Baire and analytic is Polish. Our next result hasa proof which is a minor adaptation of the proof in [BoDi]. We recall that a Hausdorfftopological space is paracompact ([Eng, Ch. 5], or [Kel, Ch. 6], especially Problem Y) ifevery open cover has a locally finite open refinement and that (i) Lindelöf spaces and (ii)metrizable spaces are paracompact. Paracompact spaces are normal, hence topological