Some results on Aronszajn trees and lines
by
David Lutzer
Mathematics Department, College of William and Mary
Coauthors: Will Funk, Vanderbilt University

The talk describes results proved by Will Funk and the speaker concerning Aronszajn lines and branch spaces of Aronszajn trees. Some of the results must be known, but we have not been able to find them in the literature. As an example of our results, we prove that if T is an Aronszajn tree with a family of node orderings, then the resulting branch space (B, < _B) satisfies: (a) B with the open interval topology is Lindelöf, first-countable, and hereditarily paracompact, and is never metrizable; (b) if T does not contain any Souslin subtree, then B is not perfect; (c) the linearly ordered set (B, < _B) contains an order copy of an uncountable set of real numbers and is therefore not an Aronszajn line; (d) However, B contains a dense subset that is order isomorphic to an Aronszajn line. A key lemma in our work is that if A is an uncountable antichain in an Aronszajn tree T and if \beta < \omega₁, then the set

S = {t in T: \textthere exists at in A with t <= T at }

is an Aronszajn tree and there is a subset C of B with cardinality 2^\omega such that each c in C is a subset of S, and the intersection of c and A is not empty, and each member of C has height > \beta.

Time permitting, we will discuss other properties of branch spaces of more general trees, such as the existence of \sigma-disjoint \pi-bases for the open interval topology of the branch space.

Date received: February 21, 2003