example of Aronszajn tree
Construction 1: If is a singular cardinal then there is a construction of a -Aronszajn tree (http://planetmath.org/KappaAronszjanTree). Let with be a sequence cofinal in . Then consider the tree where with iff and .
Note that this is similar to (indeed, a subtree of) the construction given for a tree with no cofinal branches. It consists of disjoint branches, with the -th branch of height . Since , every level has fewer than elements, and since the sequence is cofinal in , must have height and cardinality .
Construction 2: We can construct an Aronszajn tree out of the compact subsets of . will be defined by iff is an end-extension of . That is, and if and then .
Let . Given a level , let . That is, for every element in and every rational number larger than any element of , is an element of . If is a limit ordinal then each element of is the union of some branch in .
We can show by induction that for each . For the case, has only one element. If then . If is a limit ordinal then is a countable union of countable sets, and therefore itself countable. Therefore there are a countable number of branches, so is also countable. So has countable levels.
Suppose has an uncountable branch, . Then for any , . Then for each , there is some such that is greater than any element of . Then is an uncountable increasing sequence of rational numbers. Since the rational numbers are countable, there is no such sequence, so has no uncountable branch, and is therefore Aronszajn.
|Title||example of Aronszajn tree|
|Date of creation||2013-03-22 12:52:39|
|Last modified on||2013-03-22 12:52:39|
|Last modified by||Henry (455)|