example of Aronszajn tree

Construction 1: If κ is a singular cardinal then there is a construction of a κ-Aronszajn tree (http://planetmath.org/KappaAronszjanTree). Let kββ<ι with ι<κ be a sequenceMathworldPlanetmath cofinal in κ. Then consider the tree where T={(α,kβ)α<kββ<ι} with (α1,kβ1)<T(α2,kβ2) iff α1<α2 and kβ1=kβ2.

Note that this is similar to (indeed, a subtree of) the construction given for a tree with no cofinal branches. It consists of ι disjoint branches, with the β-th branch of height kβ. Since ι<κ, every level has fewer than κ elementsMathworldMathworld, and since the sequence is cofinal in κ, T must have height and cardinality κ.

Construction 2: We can construct an Aronszajn tree out of the compact subsets of +. <T will be defined by x<Ty iff y is an end-extension of x. That is, xy and if ryx and sx then s<r.

Let T0={[0]}. Given a level Tα, let Tα+1={x{q}xTαq>maxx}. That is, for every element x in Tα and every rational numberPlanetmathPlanetmathPlanetmath q larger than any element of x, x{q} is an element of Tα+1. If α<ω1 is a limit ordinalMathworldPlanetmath then each element of Tα is the union of some branch in T(α).

We can show by inductionMathworldPlanetmath that |Tα|<ω1 for each α<ω1. For the case, T0 has only one element. If |Tα|<ω1 then |Tα+1|=|Tα|||=|Tα|ω=ω<ω1. If α<ω1 is a limit ordinal then T(α) is a countableMathworldPlanetmath union of countable sets, and therefore itself countable. Therefore there are a countable number of branches, so Tα is also countable. So T has countable levels.

Suppose T has an uncountable branch, B=b0,b1,. Then for any i<j<ω1, bibj. Then for each i, there is some xibi+1bi such that xi is greater than any element of bi. Then x0,x1, is an uncountable increasing sequence of rational numbers. Since the rational numbers are countable, there is no such sequence, so T has no uncountable branch, and is therefore Aronszajn.

Title example of Aronszajn tree
Canonical name ExampleOfAronszajnTree
Date of creation 2013-03-22 12:52:39
Last modified on 2013-03-22 12:52:39
Owner Henry (455)
Last modified by Henry (455)
Numerical id 5
Author Henry (455)
Entry type Example
Classification msc 03E05
Classification msc 05C05