Normed versus topological groups: Dichotomy and duality

Normed groups 125The corresponding pseudo-metric isd(x, y) := lim d G (x n , y n ) = d G (Lx, Ly).We may take equivalence of sequences with identical limit; then C(G) ∼ becomes a normedgroup (cf. Th. 3.38). However, in our theorem below we do not wish to refer to such anequivalence.Definitions. For a family F of functions from G to H, we denote by F(T ) the family{f|T : f ∈ F} of functions in F restricted to T ⊆ G. Let us denote a convergent sequencewith limit x 0 , by {x n } → x 0 . We say the property Q of functions (property being regardedset-theoretically, i.e. as a family of functions from G to H) is sequential on T iff ∈ Q iff (∀{x n : n > 0} ⊆ T )[({x n } → x 0 ) =⇒ f|{x n : n > 0} ∈ Q({x n : n > 0})].If we further require the limit point to be enumerated in the sequence, we call Q completelysequential on T iff ∈ Q iff (∀{x n } ⊆ T )[({x n } → x 0 ) =⇒ f|{x n } ∈ Q({x n })].Our interest rests on properties that are completely sequential; our theorem below containsa condition referring to completely sequential properties, that is, the condition isrequired to hold on convergent sequences with limit included (so on a compact set), ratherthan on arbitrary sequences.Note that if Q is (completely) sequential then f|{x n } ∈ Q({x n }) iff f|{x n : n ∈ M} ∈Q({x n : n ∈ M}), for every infinite M.Definition. Let h : G → H, with G, H metric groups. Say that a sequence u = {u n } isQ-good for h ifh|{u n } ∈ Q|{u n },and putG hQ = {u : h|{u n } ∈ Q|{u n }}.If Q is completely sequential, then u is Q-good for h iff every subsequence of u is Q-goodfor h, so that G hQ is a G-ideal iff it is a complete G-ideal. One then has:Lemma 11.23. If Q is completely sequential and F preserves Q under shift and multiplicationand division on compacts, then G hQ for h ∈ F is a G-ideal.Theorem 11.24 (Analytic Automaticity Theorem - combinatorial form). Suppose thatfunctions of F having Q on G have P on G, where Q is a property of functions from Gto H that is completely sequential on G.Suppose that, for all h ∈ F, G hQ , the family of Q-good sequences is a G-ideal. Then, forany analytic set T spanning G, functions of F having Q on T have P on G.