By Rosen J. D.

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The theorem is also true for the Π-side and Δ-side of the hierarchies. 5). A basic fact for the eﬀective Borel hierarchy is that for each 1 ≤ α < ω1CK there are Σ0α sets which are not Π0α . This also relativizes. 1 (a) Show that m 1 , m2 2 = (m1 + m2 )(m1 + m2 + 1) + m2 2 is a computable bijection from ω 2 onto ω. © 2009 by Taylor & Francis Group, LLC 28 Invariant Descriptive Set Theory (b) Deﬁne · · · n for n > 2 by induction: m 1 , . . , mn Show that · · · (c) Deﬁne · <ω s = Show that · n n = m1 , .

Proof. 2 (d). To see that (ii) ⇒ (iii), let H be a Gδ subgroup of G. Then the closure of H, H, is a closed subgroup of H, and H is a dense Gδ in H. If H = H there is at least one coset gH = H in H. In H the coset gH is also a dense Gδ . 2 (b) that H ∩ gH = ∅, a contradiction. Hence H = H and so H is closed. 2 Let G be a Polish group with compatible left-invariant metric d. Then D(h, k) = d(h, k) + d(h−1 , k −1 ) is a compatible complete metric for G. Proof. 1 and therefore G = G. This shows that D is a compatible complete metric on G.

Let (Us )s∈ω<ω be a sequence of nonempty open sets in X such that, for any s ∈ ω <ω , (i) diam(Us ) ≤ 2−lh(s) ; (ii) for any n ∈ ω, Us (iii) U∅ = X; Us = n n∈ω ⊆ Us ; Us n. 7. Let Vs = π(Us ). Since π is open, each Vs is open in Y . Now let dY be a compatible metric on Y . Let Yˆ be the dY -completion of Y . Let Ns be the interior of the closure of Vs in Yˆ . Then it is easy to see that Ns ∩ Y = Vs . Also note that V∅ = Y and N∅ = Yˆ . We deﬁne a sequence (Ms ) of open sets in Yˆ such that, for any s ∈ ω <ω , (a) Ms ⊆ Ns ; (b) for any n ∈ ω, Ms n ⊆ Ms ; (c) for any y ∈ Yˆ there are only ﬁnitely many n with y ∈ Ms (d) Ms ∩ n Ms n = Ms ∩ n Ns n; n.

### [Article] Generalized polynomial identities of finite dimensional central simple algebras by Rosen J. D.

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