Normed versus topological groups: Dichotomy and duality

60 N. H. Bingham and A. J. OstaszewskiTheorem 3.45. In a topologically complete, separable, connected normed group X, if thetopological centre is non-meagre, then X is a topological group.Proof. The centre Z Γ is a closed, hence Baire, subgroup. If it is non-meagre, by Th. 6.13it is clopen and hence the whole of X (by connectedness). Again by Th. 3.4, X is atopological group.Remark. Suppose the normed group X is topologically complete and connected. Underthe circumstances, by the Squared Pettis Theorem (Th. 5.8), since Z Γ is closed and soBaire, if non-meagre it contains e X as an interior point of (Z Γ Z −1Γ)2 ; then Z Γ generatesthe whole of X. But as Z Γ is only a semigroup, we cannot deduce that X is a topologicalgroup.We now focus on conditions which yield ‘topological group’ behaviour at least ‘somewhere’.Our analysis via ‘oscillation’ sharpens Montgomery’s result concerning ‘separateimplies joint continuity’.A semitopological metric group X is a group with a metric that is not necessarilyinvariant but with right-shifts ρ y (x) = xy and left-shifts λ x (y) = xy continuous (so thatmultiplication is separately continuous). Montgomery [Mon2] proves that, in a semitopologicalmetric group, joint continuity is implied by completeness. From our perspective,we may disaggregate his result into three steps: a simple initial observation, a categoryargument (Prop. 3.46), and an appeal to oscillation. For a general metric d whichdefines the context of the first of these, we must interpret ||z|| as d(z, e) and Ω(ε) as{t : (lim δ↘0 sup ||z||≤δ d(tz, t)) < ε}. The latter set refers to left shifts, so the language ofthe initial observation corresponds to left-shift continuity.Initial Observation. In a Baire, left topological (in particular a semitopological) metricgroup, for each non-empty open set W and ε > 0, the set Ω(ε) ∩ W is non-meagre.Proof. Let ε > 0. On taking d in place of d R , this follows from (cover), since for t ∈ W,λ t is continuous at e and so there is δ > 0 such that t ∈ Λ δ (ε/2) ∩ W ⊆ Ω(ε) ∩ W . Thelatter set is thus non-empty and open, so non-meagre.The rest of his argument, using a general metric d, relies on the weaker property embodiedin the Initial Observation, that each set Ω(ε) is non-meagre in any neighbourhood.So we may interpret his arguments in a normed group context to yield two interestingresults. (The first may be viewed as defining a ‘local metric admissability condition’,compare Prop. 2.14 and the ‘uniform continuity’ of Lemma 3.5.) In Th. 3.46 below weare able to relax the hypothesis of Montgomery’s Theorem (Th. 3.47).