Topology Proceedings 28 No. 1 (2004), pp. 113-132

A family of trees with no uncountable branches

Mirna Dzamonja and Jouko Väänänen

We construct a family of 2^À₁ trees of size À₁ and no uncountable branches that in a certain way codes all w₁-sequences of infinite subsets of w. This coding allows us to conclude that in the presence of the club guessing between À₁ and À₀, these trees are pairwise very different. In such circumstances we can also conclude that the universality number of the ordered class of trees of size À₁ with no uncountable branches under "metric-preserving" reductions must be at least the continuum. From the topological point of view, the above results show that under the same assumptions there are 2^À₁ pairwise non-isometrically embeddable first countable w₁-metric spaces with a strong non-ccc property, and that their universality number under isometric embeddings is at least the continuum. Without the non-ccc requirement, a family of 2^À₁ pairwise non-isometrically embeddable first countable w₁-metric spaces exists in ZFC by an earlier result of S. Todorcevi\'c. The set-theoretic assumptions mentioned above are satisfied in many natural models of set theory (such as the ones obtained after forcing by a ccc forcing over a model of ¨). We use a similar method to discuss trees of size k with no uncountable branches, for any regular uncountable k.

Keywords: club guessing, w₁-metric, trees

Mathematics Subject Classification: 03E04 54E99

Document formats: AtlasImage format; PostScript file 269.5 Kb; PDF file 246.1 Kb

Topology Proceedings, Volume 28, No. 1 (2004)
Subscription information