Displaying similar documents to “A non-special ω 2 -tree with special ω 1 -subtrees”

Rudin's Dowker space in the extension with a Suslin tree

Teruyuki Yorioka (2008)

Fundamenta Mathematicae

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We introduce a generalization of a Dowker space constructed from a Suslin tree by Mary Ellen Rudin, and the rectangle refining property for forcing notions, which modifies the one for partitions due to Paul B. Larson and Stevo Todorčević and is stronger than the countable chain condition. It is proved that Martin's Axiom for forcing notions with the rectangle refining property implies that every generalized Rudin space constructed from Aronszajn trees is non-Dowker, and that the same...

Planting Kurepa trees and killing Jech-Кunen trees in a model by using one inaccessible cardinal

Saharon Shelah, R. Jin (1992)

Fundamenta Mathematicae

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By an ω 1 - tree we mean a tree of power ω 1 and height ω 1 . Under CH and 2 ω 1 > ω 2 we call an ω 1 -tree a Jech-Kunen tree if it has κ-many branches for some κ strictly between ω 1 and 2 ω 1 . In this paper we prove that, assuming the existence of one inaccessible cardinal, (1) it is consistent with CH plus 2 ω 1 > ω 2 that there exist Kurepa trees and there are no Jech-Kunen trees, which answers a question of [Ji2], (2) it is consistent with CH plus 2 ω 1 = ω 4 that there only exist Kurepa trees with ω 3 -many branches, which answers...

Pressing Down Lemma for λ -trees and its applications

Hui Li, Liang-Xue Peng (2013)

Czechoslovak Mathematical Journal

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For any ordinal λ of uncountable cofinality, a λ -tree is a tree T of height λ such that | T α | < cf ( λ ) for each α < λ , where T α = { x ∈ T : ht ( x ) = α } . In this note we get a Pressing Down Lemma for λ -trees and discuss some of its applications. We show that if η is an uncountable ordinal and T is a Hausdorff tree of height η such that | T α | ≤ ω for each α < η , then the tree T is collectionwise Hausdorff if and only if for each antichain C ⊂ T and for each limit ordinal α ≤ η with cf ( α ) > ω , { ht ( c ) : c ∈ C } ∩ α is not stationary in α . In the last part of this note, we investigate...