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

Akira Iwasa; Peter J. Nyikos

Commentationes Mathematicae Universitatis Carolinae (2006)

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

Abstract

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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 ♦ * , which holds in Gödel’s Constructible Universe.

How to cite

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Iwasa, 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

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  1. Devlin K.J., Shelah S., Souslin properties and tree topologies, Proc. London Math. Soc. (3) 39 (1979), 2 237-252. (1979) Zbl0432.54029MR0548979
  2. Iwasa A., Metrizability of trees, doctoral dissertation, Department of Mathematics, University of South Carolina, 2001. 
  3. Kunen K., Set Theory: An Introduction to Independence Proofs, North-Holland, Amsterdam, 1980. Zbl0534.03026MR0597342
  4. 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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