Normed versus topological groups: Dichotomy and duality

Normed groups 73Returning to the critical notion of almost completeness, we note that A almost completeis Baire resp. measurable. A bounded non-null measurable subset A is almost complete:for each ε > 0 there is a compact (so G δ ) subset K with |A\K| < ε, so we maytake N = A\K. Likewise a Baire non-meagre set in a complete metric space is almostcomplete – this is in effect a restatement of Baire’s Theorem:Theorem 5.2 (Baire’s Theorem – almost completeness of Baire sets). For a completelymetrizable space X and A ⊆ X Baire non-meagre, there is a meagre set M such thatA\M is completely metrizable and so A is almost complete.Hence, in a metrizable almost complete space a subset B is Baire iff the subspace B isalmost complete.Proof. For A ⊆ X Baire non-meagre we have A ∪ M 1 = U\M 0 with M i meagre andU a non-empty open set. Now M 0 = ⋃ n∈ω N n with N n nowhere dense; the closureF n := ¯N n is also nowhere dense (and the complement E n = X\F n is dense, open).The set M ′ 0 = ⋃ n∈ω F n is also meagre, so A 0 := U\M ′ 0 = ⋂ n∈ω U ∩ E n ⊆ A. TakingG n := U ∩ E n , we see that A 0 is completely metrizable.If X is almost complete, then any subspace of X that is almost complete is a Baire set,since an absolute G δ has the Baire property in X. As to the converse, for a Baire setB ⊆ X with X almost complete, write X = H X ∪ N X with N X meagre and H X anabsolute G δ and B = (U\M B ) ∪ N B with U open and M B , N B meagre. We have just seenthat without loss of generality M B may be taken to be a meagre F σ subset of U (otherwisechoose F B a meagre F σ containing M B and let F B and N B ∪ (F B \M B ) replace M B andN B respectively). Intersecting the representations of X and B, one has B = H B ∪ N ′ Bfor H B := H X ∩ (U\F B ), an absolute G δ , and some meagre N ′ B ⊆ N B ∪ N X . So, B isalmost complete.Th. 5.2 says that, in a complete space, a set which is almost open is almost complete.More generally, even if the space is not complete, any non-meagre separable analytic set(for definition of which see Section 11) is almost complete – a result observed by S. Levi in[Levi]. (More in fact is true – see [Ost-AH] Cor. 2 and [Ost-AB].) In an almost completespace the distinction between the two notions of Baire property and Baire subspace isblurred, the two being indistinguishable. Almost completely metrizable spaces may becharacterized in a useful fashion by reference to a less demanding absoluteness conditionthan topological completeness (we recall the latter is equivalent to being an absolute G δ– see above). It may be shown that a non-meagre normed group is almost complete iff itis almost absolutely analytic (see [Ost-AB],[Ost-LB3]).The KBD Theorem is a generic assertion about embedding into target sets. We addressfirst the source of this genericity, which is that a property inheritable by supersets eitherholds generically or fails outright. This is now made precise.