Rod Downey and Antonio Montalbán
نویسنده
چکیده
This theorem will be used in both constructions, the one of a non-slender thin class, and the one of a slender thin class. The proof starts developing ideas the will be used in both of those constructions. In the case when [T ] is perfect, we will define two tree-embeddings f, r : 2 → S satisfying that for every σ ∈ 2, (fr1) f(σ) ⊆ r(σ), f(σ0) = r(σ)0, f(σ1) = r(σ)1, and (fr2) [Sf(σ)] = [Sr(σ)] It follows that [S] = [image(f)] = [image(r)], and hence that [S] is perfect. There are two types of requirements: the thinness requirements Te : Fe ⊆ S ⇒ ∃C ⊆ 2 clopen ([Fe] = [S] ∩ C), and the finiteness requirements Fe : [Tte ] is isolated ⇒ [S] is finite. where {t0, t1, ...} is an enumeration of T . These requirements are subdivided even further. Each requirement Te is divided into 2 2e sub-requirements Tσ, one for each σ ∈ 2 . Tσ : either [(Fe)f(σ)] = [Sf(σ)], or [Fe] ∩ [Sf(σ)] = ∅. Note that if all the requirements Tσ for σ ∈ 2 are satisfied, then so is Te by letting C = ⋃ {[f(σ)] : σ ∈ 2 & [(Fe)f(σ)] = [Sf(σ)]}. The strategy of Tσ is roughly the following. If Tσ sees the opportunity to define r(σ) 6∈ Fe, it will do it guaranteeing that Fe ∩ Sr(σ) = ∅. If such an opportunity never appears, it is because (Fe)f(σ) = Sf(σ).
منابع مشابه
Slender Classes . Rod Downey And
A Π1 class P is called thin if, given a subclass P ′ of P there is a clopen C with P ′ = P ∩C. Cholak, Coles, Downey and Herrmann [7] proved that a Π1 class P is thin if and only if its lattice of subclasses forms a Boolean algebra. Those authors also proved that if this boolean algebra is the free Boolean algebra, then all such think classes are automorphic in the lattice of Π1 classes under i...
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