This is a short overview of a talk given by Stefan Geschke on November 21, 2008. Stefan’s topic, Cofinalities of algebraic structures and coinitialities of topological spaces, very quickly connects set theory with other areas, and leads to well-known open problems. In what follows, compact always includes Hausdorff. Most of the arguments I show below are really only quick sketches rather than complete proofs. Any mistakes or inaccuracies are of course my doing rather than Stefan’s, and I would be grateful for comments, corrections, etc.
Definition. Let be a (first order) structure in a countable language. Write for the smallest such that for a strictly increasing union of proper substructures.
Since the structures need to be proper, is not defined if is finite. It may also fail to exist if is countable, but it is defined if is uncountable. Moreover, if exists, then
- , and
- is a regular cardinal.
Example 1. Groups can have arbitrarily large cofinality. This is not entirely trivial, as the sets may have size .
Question 1. Is every regular cardinal realized this way?