Normed versus topological groups: Dichotomy and duality

88 N. H. Bingham and A. J. Ostaszewskiversions: a normed topological group version, immediately following, and a normed groupversion given in Theorem 6.13; the proofs are rather different.Theorem 6.11 (Subgroup Dichotomy Theorem – normed topological groups, Banach-KuratowskiTheorem – [Ban-G, Satz 1], [Kur-1, Ch. VI. 13. XII]; cf. [Kel, Ch. 6 Pblm P] ; cf.[BGT, Cor. 1.1.4] and also [BCS] and [Be] for the measure variant). Let X be a normedtopological group which is non-meagre and A any Baire subgroup. Then A is either meagreor clopen in X.Proof. Suppose that A is non-meagre. We show that e is an interior point of A, fromwhich it follows that A is open. Suppose otherwise. Then there is a sequence z n → e withz n ∈ B 1/n (e)\A. Now for some a ∈ A and infinite M we have z n a ∈ A for all n ∈ M. ButA is a subgroup, hence z n = z n aa −1 ∈ A for n ∈ M, a contradiction.Now suppose that A is not closed. Let a n be a sequence in A with limit x. Then a n x −1 →e. Now for some a ∈ A and infinite M we have z n x −1 a ∈ A for all n ∈ M. But Ais a subgroup, so zn −1 and a −1 are in A and hence, for all n ∈ M, we have x −1 =z −1nz n x −1 aa −1 ∈ A. Hence x ∈ A, as A is a subgroup.Remark. Banach’s proof is purely topological, so applies to topological groups (eventhough originally stated for metric groups), and relies on the mapping x → ax beinga homeomorphism, likewise Kuratowski’s proof, which proceeds via another dichotomyas detailed below. We refer to McShane’s proof, cited below, as it yields a slightly moregeneral version.Theorem 6.12 (Kuratowski-McShane Dichotomy – [Kur-B], [Kur-1], [McSh, Cor. 1] ).Suppose H ⊆ Auth(X) acts transitively on the topological space X, and Z ⊆ X is Baireand has the property that for each h ∈ HZ = h(Z) or Z ∩ h(Z) = ∅,i.e. under each h ∈ H, either Z is invariant or Z and its image are disjoint. Then, eitherZ is meagre or it is clopen.Theorem 6.13 (Subgroup Dichotomy Theorem – normed groups). In a normed groupX, Baire under its norm topology, a Baire non-meagre subgroup is clopen.Proof. We work under the right norm topology and denote the subgroup in question S .Let H := {ρ x : x ∈ X} ⊆ Auth(X). Then as S is a subgroup, for x ∈ S, ρ x (S) = S, and,for x /∈ S, ρ x (S) ∩ S = ∅. Hence, by the Kuratowski-McShane Dichotomy (Th. 6.12), asS is non-meagre, it is clopen.