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Pressing Down Lemma for λ -trees and its applications

Hui Li, Liang-Xue Peng (2013)

Czechoslovak Mathematical Journal

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 some...

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