By Su Gao

*Presents effects from a truly lively zone of Research*

Exploring an lively zone of arithmetic that reports the complexity of equivalence kin and type difficulties, **Invariant Descriptive Set concept provides an creation to the elemental ideas, equipment, and result of this thought. It brings jointly suggestions from numerous parts of arithmetic, similar to algebra, topology, and common sense, that have different functions to different fields.**

After reviewing classical and potent descriptive set concept, the textual content reviews Polish teams and their activities. It then covers Borel reducibility effects on Borel, orbit, and normal definable equivalence family members. the writer additionally presents proofs for various primary effects, reminiscent of the Glimm–Effros dichotomy, the Burgess trichotomy theorem, and the Hjorth turbulence theorem. the following half describes connections with the countable version concept of infinitary good judgment, in addition to Scott research and the isomorphism relation on ordinary periods of countable versions, resembling graphs, bushes, and teams. The e-book concludes with functions to category difficulties and lots of benchmark equivalence kin.

By illustrating the relevance of invariant descriptive set thought to different fields of arithmetic, this self-contained booklet encourages readers to additional discover this very energetic region of study.

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**Extra resources for Invariant descriptive set theory**

**Example text**

We show the unrelativized version only, and for this it suﬃces to show that if y ∈ Δ11 and y ∈ WO, then ot(

Show that rk(T ) < ω1CK . 6 29 Analytic sets and Σ11 sets In this section we continue to consider subsets of product spaces X = ω k × (ω ω )l . Recall that a subset A ⊆ X is analytic (or Σ11 ) if there is a closed subset C of X × ω ω such that, for all x ∈ X, x ∈ A ⇐⇒ ∃y ∈ ω ω (x, y) ∈ C. Similarly, we deﬁne that A ⊆ X is Σ11 if there is a Π01 set C of X × ω ω such that x ∈ A ⇐⇒ ∃y ∈ ω ω (x, y) ∈ C. In addition, A ⊆ X is coanalytic (or Π11 ) if X − A is analytic. A is Δ11 if it is both analytic and coanalytic.

7 Let X be a Polish space. Show that the union operator ∪ : F (X) × F (X) → F (X) → F1 ∪ F2 (F1 , F2 ) is Borel. 5 23 The eﬀective hierarchy The introduction of the notion of computability into descriptive set theory profoundly changed the subject. The resulting theory is often referred to as eﬀective descriptive set theory, and concepts and results obtained in the precomputability theory era is now called classical descriptive set theory. The eﬀective theory is more fundamental since it can be developed independently and it covers the classical theory completely.