Tukey classes of local bases in compacta
by
David Milovich
University of Wisconsin-Madison

I will survey some recent results about the Tukey classes of local bases at points in compacta (ordered by containment). In any compactum with all points' pi-character at least kappa, the set of finite subsets of kappa, ordered by inclusion, is Tukey reducible to some local base, which implies that every compactum has a point with a countable local pi-base or a point with no well-quasiordered local base.

For the Cech-Stone remainder of omega, there are several independence results about the spectrum of Tukey classes of its local bases.

For all points p in a dyadic compactum X, local bases at p are Tukey equivalent to the set of finite subsets of the character of p. The last statement is also true of all known examples of homogeneous compacta.

Date received: February 18, 2007