Sequential compactness (and lack thereof) in topological spaces
by
Peter Nyikos
Dept. of Mathematics, University of South Carolina, Columbia, SC 29208

For once, no separation axioms are assumed. The cardinal number h can be characterized as the least height of a tree p-base for the Stone-Cech remainder of the integers. A proof of the following theorem will be outlined:

Theorem. h is the least cardinality of a countably compact space that is not sequentially compact.

The space used for one direction is a KC-space, meaning that every compact subset is closed. This is about as good as it gets: it is known that every countably compact Hausdorff space of cardinality less than the splitting number s (which is consistently greater than h) is sequentially compact. It is still unknown whether s is optimal here. Related theorems and problems will be briefly discussed.

Date received: February 28, 2006