# A non-metrizable collectionwise Hausdorff tree with no uncountable chains and no Aronszajn subtrees

Commentationes Mathematicae Universitatis Carolinae (2006)

- Volume: 47, Issue: 3, page 515-523
- ISSN: 0010-2628

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topIwasa, Akira, and Nyikos, Peter J.. "A non-metrizable collectionwise Hausdorff tree with no uncountable chains and no Aronszajn subtrees." Commentationes Mathematicae Universitatis Carolinae 47.3 (2006): 515-523. <http://eudml.org/doc/249866>.

@article{Iwasa2006,

abstract = {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.},

author = {Iwasa, Akira, Nyikos, Peter J.},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {tree; collectionwise Hausdorff; metrizable; Aronszajn tree; tree; collectionwise Hausdorff; metrizable; Aronszajn tree},

language = {eng},

number = {3},

pages = {515-523},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {A non-metrizable collectionwise Hausdorff tree with no uncountable chains and no Aronszajn subtrees},

url = {http://eudml.org/doc/249866},

volume = {47},

year = {2006},

}

TY - JOUR

AU - Iwasa, Akira

AU - Nyikos, Peter J.

TI - A non-metrizable collectionwise Hausdorff tree with no uncountable chains and no Aronszajn subtrees

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2006

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 47

IS - 3

SP - 515

EP - 523

AB - 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.

LA - eng

KW - tree; collectionwise Hausdorff; metrizable; Aronszajn tree; tree; collectionwise Hausdorff; metrizable; Aronszajn tree

UR - http://eudml.org/doc/249866

ER -

## References

top- Devlin K.J., Shelah S., Souslin properties and tree topologies, Proc. London Math. Soc. (3) 39 (1979), 2 237-252. (1979) Zbl0432.54029MR0548979
- Iwasa A., Metrizability of trees, doctoral dissertation, Department of Mathematics, University of South Carolina, 2001.
- Kunen K., Set Theory: An Introduction to Independence Proofs, North-Holland, Amsterdam, 1980. Zbl0534.03026MR0597342
- Nyikos P.J., Metrizability, monotone normality, and other strong properties in trees, Topology Appl. 98 (1999), 269-290. (1999) Zbl0969.54026MR1720006

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