Normed versus topological groups: Dichotomy and duality

116 N. H. Bingham and A. J. OstaszewskiRemark. Hansell’s condition, requiring the function f to be σ-discrete, is implied by fbeing analytic when X is absolutely analytic (i.e. Souslin-F(X) in any complete metricspace X into which it embeds). Frankiewicz and Kunen in [FrKu] study the consistencyrelative to ZFC of the existence of a Baire function failing to have Baire continuity.The following result provides a criterion for verifying that f is Baire.Theorem 11.9 (Souslin criterion - for Baire functions). Let X and Y be Hausdorff topologicalspaces with Y a K-analytic space. If f : X → Y has Souslin-F(X × Y ) graph,then f is Baire.Proof. Let G ⊆ X × Y be the graph of f which is Souslin-F(X × Y ). For F closed in Y,we havef −1 (F ) = pr X [G ∩ (X × F )],which, by the Intersection Theorem (Th. 11.3), is the projection of a Souslin-F(X × Y )set. By the Projection Theorem (Th. 11.4), f −1 (F ) is Souslin-F(X). Closed sets have theBaire property by definition, so by Nikodym’s Theorem f −1 (F ) has the Baire property.We note that in the realm of separable metric spaces, a surjective map f with analyticgraph is in fact Borel (since for U open f −1 (U) and f −1 (Y \U) are complementaryanalytic and so Borel sets, by Souslin’s Theorem (see [Jay-Rog] Th.1.4.1); for the nonseparablecase compare [Han-71, Th. 4.6(a)]).Before stating our next theorem we recall a classical result in a sharper form. We aregrateful to the referee for the statement and proof of this result in the topological groupsetting, here amended to the normed group setting.Theorem 11.10 (Banach-Mehdi Continuity Theorem – [Ban-T, 1.3.4, p. 40], [Meh], [HJ,Th. 2.2.12 p. 348], or [BOst-TRII]). A Baire-continuous homomorphism f : X → Ybetween normed groups, with X Baire in the norm topology, is continuous. In particularthis is so for f Borel-measurable and Y separable.Proof. We work with the right norm topologies without loss of generality, since inversionis a homomorphism and also an isometry from the left to the right norm topology (Prop.2.5). We claim that it is enough to prove the following: for any non-empty open G inX and any ε > 0 there is a non-empty open V ⊆ G with diam(f(V )) < ε. Indeed theclaim implies that for each n ∈ N the set W n := ⋃ {V : diam(f(V )) < 1/n and V isopen and non-empty} is dense and open in X. Hence, as X is Baire, the intersection⋂n∈ W n is a non-empty set containing continuity points of f; but f is a homomorphismso is continuous everywhere.Now fix G non-empty and open and ε > 0. As f is Baire-continuous, it is continuous when