By Krzysztof Ciesielski

The following the authors formulate and discover a brand new axiom of set concept, CPA, the overlaying estate Axiom. CPA is in keeping with the standard ZFC axioms, certainly it truly is actual within the iterated Sacks version and truly captures the combinatorial middle of this version. A plethora of effects identified to be actual within the Sacks version simply persist with from CPA. exchanging iterated forcing arguments with deductions from CPA simplifies proofs, offers deeper perception, and results in new effects. One could say that CPA is identical in nature to Martin's axiom, as either catch the essence of the versions of ZFC within which they carry. The exposition is a self contained and there are traditional purposes to genuine research and topology. Researchers that use set concept of their paintings will locate a lot of curiosity during this publication.

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Bartoszy´ nski [4, thm. 9]) cof(N ) = min |F|: F ⊂ CH & F = ωω . 2 The family CH = {X ⊂ ω ω : X ⊂ T for some T ∈ CH } is ω Fcube -dense in Perf(ω ). Proof. Let f : Cω → ω ω be a continuous function. 4) it is enough ∗ to ﬁnd a perfect cube C in Cω such that f [C] ∈ CH . Construct, by induction on n < ω, the families {Esi : s ∈ 2n & i < ω} of nonempty clopen subsets of C such that, for every n < ω and s, t ∈ 2n , (i) Esi = Eti for every n ≤ i < ω; i i (ii) Esˆ0 and Esˆ1 are disjoint subsets of Esi for every i < n + 1; (iii) for every si ∈ 2n : i < ω 2 f (x0 ) 2(n+1) = f (x1 ) 2(n+1) 2 for every x0 , x1 ∈ Esi .

If bσ > 0, then bσˆ0 > 0 and bσˆ1 > 0. If λ = dom(σ) is a limit ordinal, then bσ = ξ<λ bσ ξ . Let T = {s ∈ 2<ω1 : bs > 0}. Then T is a subtree of 2<ω1 ; its levels determine antichains in B, so they are countable. First assume that T has a countable height. Then T itself is countable. Let B0 be the smallest complete subalgebra of B containing {bσ : σ ∈ T } and notice that B0 is atomless. Indeed, if there were an atom a in B0 , S then S = {σ ∈ T : a ≤ bσ } would be a branch in T so that δ = would belong to 2<ω1 .

17(a)] that every compact set C ∈ Perf(R) contains a dense Gδ subset that belongs to Zp . 5, the elements of Perf(R) ∩ Zp are Fcube -dense in Perf(R). 4 CPAcube implies that cov(B1 , Perf(R)) = cov(Zp ) = ω1 . 3 cof(N ) = ω1 Next, following the argument of K. Ciesielski and J. Pawlikowski from [39], we show that CPAcube implies that cof(N ) = ω1 . , [4]) are equal to ω1 . Let CH be the family of all subsets n<ω Tn of ω ω such that Tn ∈ [ω]≤n+1 for all n < ω. We will use the following characterization.