Pressing Down Lemma for $\lambda$-trees and its applications

Li, Hui, and Peng, Liang-Xue. "Pressing Down Lemma for $\lambda $-trees and its applications." Czechoslovak Mathematical Journal 63.3 (2013): 763-775. <http://eudml.org/doc/260616>.

@article{Li2013,
abstract = {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 \}=\lbrace x\in T\colon \{\rm ht\}(x)=\alpha \rbrace $. 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 \}|\le \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 \le \eta $ with $\{\rm cf\}(\alpha )>\omega $, $\lbrace \{\rm ht\}(c)\colon c\in C\rbrace \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.},
author = {Li, Hui, Peng, Liang-Xue},
journal = {Czechoslovak Mathematical Journal},
keywords = {tree; $D$-space; $\lambda $-tree; property $\gamma $; collectionwise Hausdorff; tree; -space; -tree; property ; collectionwise Hausdorff},
language = {eng},
number = {3},
pages = {763-775},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Pressing Down Lemma for $\lambda $-trees and its applications},
url = {http://eudml.org/doc/260616},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Li, Hui
AU - Peng, Liang-Xue
TI - Pressing Down Lemma for $\lambda $-trees and its applications
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 3
SP - 763
EP - 775
AB - 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 }=\lbrace x\in T\colon {\rm ht}(x)=\alpha \rbrace $. 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 }|\le \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 \le \eta $ with ${\rm cf}(\alpha )>\omega $, $\lbrace {\rm ht}(c)\colon c\in C\rbrace \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.
LA - eng
KW - tree; $D$-space; $\lambda $-tree; property $\gamma $; collectionwise Hausdorff; tree; -space; -tree; property ; collectionwise Hausdorff
UR - http://eudml.org/doc/260616
ER -