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Keywords:
tree; $D$-space; $\lambda $-tree; property $\gamma $; collectionwise Hausdorff
tree; $D$-space; $\lambda $-tree; property $\gamma $; collectionwise Hausdorff
Summary:
For any ordinal $\lambda $ of uncountable cofinality, a $\lambda $-tree is a tree $T$ of height $\lambda $ such that $|T_{\alpha }|<{\rm cf}(\lambda )$ for each $\alpha <\lambda $, where $T_{\alpha }=\{x\in T\colon {\rm ht}(x)=\alpha \}$. In this note we get a Pressing Down Lemma for $\lambda $-trees and discuss some of its applications. We show that if $\eta $ is an uncountable ordinal and $T$ is a Hausdorff tree of height $\eta $ such that $|T_{\alpha }|\leq \omega $ for each $\alpha <\eta $, then the tree $T$ is collectionwise Hausdorff if and only if for each antichain $C\subset T$ and for each limit ordinal $\alpha \leq \eta $ with ${\rm cf}(\alpha )>\omega $, $\{{\rm ht}(c)\colon c\in C\} \cap \alpha $ is not stationary in $\alpha $. In the last part of this note, we investigate some properties of $\kappa $-trees, $\kappa $-Suslin trees and almost $\kappa $-Suslin trees, where $\kappa $ is an uncountable regular cardinal.
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