Read e-book online The Covering Property Axiom, CPA: A Combinatorial Core of PDF

By Krzysztof Ciesielski

ISBN-10: 0511217404

ISBN-13: 9780511217401

ISBN-10: 0521839203

ISBN-13: 9780521839204

The following the authors formulate and discover a brand new axiom of set concept, CPA, the masking estate Axiom. CPA is in keeping with the standard ZFC axioms, certainly it's precise within the iterated Sacks version and really captures the combinatorial middle of this version. A plethora of effects identified to be real within the Sacks version simply stick with from CPA. exchanging iterated forcing arguments with deductions from CPA simplifies proofs, offers deeper perception, and ends up in new effects. One might say that CPA is the same in nature to Martin's axiom, as either catch the essence of the types of ZFC within which they carry. The exposition is a self contained and there are typical functions to genuine research and topology. Researchers that use set conception of their paintings will locate a lot of curiosity during this e-book.

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So, E is Fcube -dense. By CPAcube , there is an E0 ⊂ E such that |E0 | ≤ ω1 and |X \ E0 | ≤ ω1 . Let P0 = {P ∈ E0 : P ⊂ B} and P = P0 ∪ {{x}: x ∈ B \ E0 }. Then P is as desired. 2 Uniformly completely Ramsey null sets Uniformly completely Ramsey null sets are small subsets of [ω]ω that are related to the Ramsey property. The notion has been formally defined by U. Darji [47], though it was already studied by F. Galvin and K. Prikry in [63]. Instead of using the original definition for this class, we will use its characterization due to A.

It is easy to see that h is a homeomorphism and that if P ⊂ Cn is a perfect cube, then h[P ] is a perfect cube. 30 1 Axiom CPAcube and its consequences: properties (A)–(E) Now, let F ⊂ Perf(X) be n-cube dense in X. To see that F is m-cube dense, take a continuous injection f : Cm → X. Then f ◦ h: Cn → X is also a continuous injection. Since F is n-cube dense, there exists a P ⊂ Cn such that P is a perfect cube and f [h[P ]] = (f ◦ h)[P ] ∈ F. But h[P ] is a perfect cube, so F is m-cube dense. 1 shows that 1-cube density does not imply 2-cube density.

So, E is Fcube -dense. By CPAcube , there is an E0 ⊂ E such that |E0 | ≤ ω1 and |X \ E0 | ≤ ω1 . Let P0 = {P ∈ E0 : P ⊂ B} and P = P0 ∪ {{x}: x ∈ B \ E0 }. Then P is as desired. 2 Uniformly completely Ramsey null sets Uniformly completely Ramsey null sets are small subsets of [ω]ω that are related to the Ramsey property. The notion has been formally defined by U. Darji [47], though it was already studied by F. Galvin and K. Prikry in [63]. Instead of using the original definition for this class, we will use its characterization due to A.

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The Covering Property Axiom, CPA: A Combinatorial Core of the Iterated Perfect Set Model by Krzysztof Ciesielski


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