Aronszajn tree

A $\kappa$ -tree (http://planetmath.org/TreeSetTheoretic) $T$ for which $|T_{\alpha}|<\kappa$ for all $\alpha<\kappa$ and which has no cofinal branches is called a $\kappa$ -Aronszajn tree. If $\kappa=\omega_{1}$ then it is referred to simply as an Aronszajn tree.

If there are no $\kappa$ -Aronszajn trees for some $\kappa$ then we say $\kappa$ has the tree property. $\omega$ has the tree property, but no singular cardinal has the tree property.

Title	Aronszajn tree
Canonical name	AronszajnTree
Date of creation	2013-03-22 12:52:34
Last modified on	2013-03-22 12:52:34
Owner	Henry (455)
Last modified by	Henry (455)
Numerical id	10
Author	Henry (455)
Entry type	Definition
Classification	msc 03E05
Classification	msc 05C05
Related topic	TreeSetTheoretic
Related topic	Antichain
Related topic	SuslinTree
Related topic	WeaklyCompactCardinalsAndTheTreeProperty
Defines	Aronszajn tree
Defines	$\kappa$ -Aronszajn tree
Defines	tree property