Countable compactness in groups, semigroups and Cartesian products, by Arthur Tomita

Topology Atlas Document # zaaa-51.htm | Production Editor: R. Flagg

Topology Atlas Invited Contributions, vol. 1, issue 2 (1996), 11.

Countable compactness in groups, semigroups and Cartesian products
by

Arthur Tomita

(University of Sao Paulo, Institute of Mathematics and Statistics, Caixa Postal 66281, 05389-970 Sao Paulo, Brazil)

Let's consider the following: Is there an example of

(a)(Wallace-1953) a countably compact both-sided cancellative semigroup which is not a topological group,

(b)(Tkachenko-1990) a countably compact free abelian group,

(c')(Comfort-1990) for which k < 2^c there exist a group whose k-th power is countably compact but whose k^+-power is not.

These are questions that could be understood by any student with some basic knowledge of topology and groups. However all the known partial solutions depend on some set theoretic assumption and the constructions are not easy and are related to each other.

The known examples:

(a) Robbie and Svetlichny (1994) under CH. We now have one under MA_countable.

(b) Tkachenko (1990) under CH. We now have one under MA(sigma-centered) whose square is not countably compact.

(c) van Douwen (1980) under MA for two groups.We showed under MA_countable that for every k IN N there exist k+1 groups whose subproduct of k many of them is countably compact but whose product of all of them is not.

(c')Hart and van Mill (1991) under MA_countable k=1. We have one group under MA_countable for k=2. Recently we showed that there are infinitely many such k's in omega.

Finally we note that an example for (a) (Tomita-1995) or (c')(Ginsburg and Saks-1974) cannot have the 2^c-th power countably compact and for (b)(Tomita-1995) cannot have the omega-th power countably compact.

Received by the editors: December 14, 1995.