By Su Gao
Presents effects from a really lively quarter of Research
Exploring an lively sector of arithmetic that stories the complexity of equivalence family and type difficulties, Invariant Descriptive Set idea provides an advent to the elemental suggestions, tools, and result of this conception. It brings jointly options from numerous parts of arithmetic, corresponding to algebra, topology, and good judgment, that have various purposes to different fields.
After reviewing classical and powerful descriptive set concept, the textual content stories Polish teams and their activities. It then covers Borel reducibility effects on Borel, orbit, and common definable equivalence family. the writer additionally presents proofs for various basic effects, corresponding to the Glimm–Effros dichotomy, the Burgess trichotomy theorem, and the Hjorth turbulence theorem. the subsequent half describes connections with the countable version idea of infinitary common sense, in addition to Scott research and the isomorphism relation on ordinary sessions of countable versions, equivalent to graphs, bushes, and teams. The e-book concludes with purposes to type difficulties and plenty of benchmark equivalence relatives.
By illustrating the relevance of invariant descriptive set idea to different fields of arithmetic, this self-contained booklet encourages readers to additional discover this very energetic zone of study.
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Additional info for Invariant descriptive set theory
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.
Invariant descriptive set theory by Su Gao