By Jenca G.

We end up that if E1 and E2 are a-complete influence algebras such that E1 is an element of E2 and E2 is an element of E1, then E1 and E2 are isomorphic.

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It follows from the deﬁnition of d∗ and the left-invariance of d that for any k there are nk and hk ∈ H such that d(hk gnk , g∞ ) < 2−k . This means −1 −1 that g∞ is an accumulation point of πH (A). Since A is closed in G/H, πH (A) −1 is closed in G, and thus g∞ ∈ πH (A), or πH (g∞ ) = Hg∞ ∈ A. −1 Conversely, let A be closed in the d∗ -topology. We show that πH (A) is −1 closed in G. For this let (gn ) be a sequence in πH (A) and assume that © 2009 by Taylor & Francis Group, LLC 48 Invariant Descriptive Set Theory gn → g∞ as n → ∞.

This ﬁnishes the proof of the claim and that d is a metric. It is clear that ρ is left-invariant and consequently so is d. © 2009 by Taylor & Francis Group, LLC 42 Invariant Descriptive Set Theory Finally we verify that d is compatible with the topology of G. Let U be open in G and g ∈ U . Then for some n ∈ N, gVn ⊆ U . We check that Bd (g, 2−n−1 ) = {h ∈ G | d(g, h) < 2−n−1 } ⊆ U . Let h ∈ Bd (g, 2−n−1 ). Then d(h, g) ∈ 2−n−1 . By the claim in the proof above ρ(g, h) ≤ 2d(g, h) < 2−n . From the deﬁnition of ρ, g −1 h ∈ Vn .

3 A real x ∈ ω ω is Δ11 if its graph {(n, m) : x(n) = m} ⊆ ω 2 is Δ11 . Show that the following are equivalent: (1) x is Δ11 ; (2) {x} is Δ11 ; (3) {x} is Σ11 . Moreover, the statement can be relativized. 8 The Gandy–Harrington topology In this section we introduce the Gandy–Harrington topology and review some facts relevant to this topic. This topic is more technical than the results reviewed in the previous sections, but is necessary in the study of equivalence relations in Part II of this book.