| Title:
|
Pressing Down Lemma for $\lambda $-trees and its applications (English) |
| Author:
|
Li, Hui |
| Author:
|
Peng, Liang-Xue |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
3 |
| Year:
|
2013 |
| Pages:
|
763-775 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
tree |
| Keyword:
|
$D$-space |
| Keyword:
|
$\lambda $-tree |
| Keyword:
|
property $\gamma $ |
| Keyword:
|
collectionwise Hausdorff |
| MSC:
|
54F05 |
| MSC:
|
54F65 |
| idZBL:
|
Zbl 06282108 |
| idMR:
|
MR3125652 |
| DOI:
|
10.1007/s10587-013-0050-0 |
| . |
| Date available:
|
2013-10-07T12:06:58Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143487 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| . |
