| Title:
|
A non-metrizable collectionwise Hausdorff tree with no uncountable chains and no Aronszajn subtrees (English) |
| Author:
|
Iwasa, Akira |
| Author:
|
Nyikos, Peter J. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
47 |
| Issue:
|
3 |
| Year:
|
2006 |
| Pages:
|
515-523 |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is independent of the usual (ZFC) axioms of set theory whether every collectionwise Hausdorff tree is either metrizable or has an uncountable chain. We show that even if we add ``or has an Aronszajn subtree,'' the statement remains ZFC-independent. This is done by constructing a tree as in the title, using the set-theoretic hypothesis $\diamondsuit^*$, which holds in Gödel's Constructible Universe. (English) |
| Keyword:
|
tree |
| Keyword:
|
collectionwise Hausdorff |
| Keyword:
|
metrizable |
| Keyword:
|
Aronszajn tree |
| MSC:
|
03E05 |
| MSC:
|
54A35 |
| MSC:
|
54E35 |
| MSC:
|
54F05 |
| idZBL:
|
Zbl 1150.54005 |
| idMR:
|
MR2281013 |
| . |
| Date available:
|
2009-05-05T16:59:12Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119612 |
| . |
| Reference:
|
[1] Devlin K.J., Shelah S.: Souslin properties and tree topologies.Proc. London Math. Soc. (3) 39 (1979), 2 237-252. Zbl 0432.54029, MR 0548979 |
| Reference:
|
[2] Iwasa A.: Metrizability of trees.doctoral dissertation, Department of Mathematics, University of South Carolina, 2001. |
| Reference:
|
[3] Kunen K.: Set Theory: An Introduction to Independence Proofs.North-Holland, Amsterdam, 1980. Zbl 0534.03026, MR 0597342 |
| Reference:
|
[4] Nyikos P.J.: Metrizability, monotone normality, and other strong properties in trees.Topology Appl. 98 (1999), 269-290. Zbl 0969.54026, MR 1720006 |
| . |
