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Article

Iwasa, Akira ; Nyikos, Peter J.
A non-metrizable collectionwise Hausdorff tree with no uncountable chains and no Aronszajn subtrees. (English). Commentationes Mathematicae Universitatis Carolinae, vol. 47 (2006), issue 3, pp. 515-523
MSC: 03E05, 54A35, 54E35, 54F05 | MR 2281013 | Zbl 1150.54005
Keywords:
tree; collectionwise Hausdorff; metrizable; Aronszajn tree
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.
References:
[1] Devlin K.J., Shelah S.: Souslin properties and tree topologies. Proc. London Math. Soc. (3) 39 (1979), 2 237-252. MR 0548979 | Zbl 0432.54029
[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. MR 0597342 | Zbl 0534.03026
[4] Nyikos P.J.: Metrizability, monotone normality, and other strong properties in trees. Topology Appl. 98 (1999), 269-290. MR 1720006 | Zbl 0969.54026
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