Compactness and the Axiom of Choice
by
Horst Herrlich
Universität Bremen

In the absence of the axiom of choice several versions of compactness (equivalent under AC) are investigated - in particular with respect to the validity of the theorems of Tychonoff and of Cech-Stone. Typical results:

1. C-compact spaces form the epireflective hull in Haus of A-compact completely regular spaces.
2. Boolean spaces form the epireflective hull in Haus of A-compact zero-dimensional spaces.
3. Equivalent are:

   1. the axiom of choice
   2. products of A-compact spaces are A-compact
   3. products of D-compact Hausdorff spaces are D-compact
   4. finite products of D-compact spaces are D-compact
   5. each closed filter is contained in a maximal one.
4. Equivalent are:

   1. the Boolean prime ideal theorem
   2. products of A-compact Hausdorff spaces are A-compact
   3. each z-filter is contained in a maximal one.
5. Equivalent are:

   1. products of B-compact spaces are B-compact
   2. either the axiom of choice holds or every ultrafilter is fixed.

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caae-36.