Countably compact groups and p-limits
by
Artur H. Tomita
Universidade de São Paulo
Coauthors: S. Garcia-Ferreira (UNAM, Mexico)

The concept of p-limit, introduced by Berstein [B] is widely used in the construction of countably compact spaces. In the same paper, Berstein also introduced p-compactness which is related to the countable compactness of products of spaces. Garcia Ferreira generalized p-compactness in [G]:

Definition 1. Let M be a non-empty subset of free ultrafilters on \omega. A topological space X is quasi M-compact if for every sequence (x_n)_{n in \omega} there exist p in M and x in X such that x is the p-limit of this sequence.

Note that the notion of countably compactness is equivalent to quasi-\omega^*-compactness and p-compactness is equivalent to quasi {p}-compactness.

Thus, quasi M-compactness can be viewed as an evaluation of how many different kinds of p-limits are required to witness the countable compactness of a space.

A notion closely related to p-compactness is almost p-compactness. Denote by T(p) the set of all ultrafilters that are Rudin-Keisler equivalent to p. A space is almost p-compact if it is quasi T(p)-compact.

Garcia Ferreira [G] showed that unlike p-compactness, almost p-compactness is not productive.

In this talk we present some examples of quasi M-compact groups which follow the joint efforts started in [GT].

The first example shows that there are almost p-compact groups which are not productive under p=c. The second example obtained via forcing, shows that some countably compact groups are not quasi M-compact if |M| < 2^c.

In [S], Saks showed that a well-known theorem in [GS] about the countable compactness of product of spaces is also consistently the best possible for topological spaces. Our third example also via forcing, shows that that theorem is also consistently the best possible for topological groups.

References

[B] A. R. Bernstein, A new kind of compactness for topological spaces, Fund. Math. 66 (1970), 185-193.

[G] S. Garcia-Ferreira, Quasi M-compact spaces, Czech. Math. J. 46 (1996), 161-177.

[GT] S. Garcia-Ferreira and A. H. Tomita, Countable compactness and p-limits, preprint.

[GS] J. Ginsburg and V. Saks, Some applications of ultrafilters in topology, Pacific J. Math. 57 (1975), 403-418.

[S] V. Saks, Ultrafilters invariants intopological spaces, Trans. Amer. Math. Soc. 241 (1978), 79-97.

Date received: March 2, 2001

Copyright © 2001 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 # cafw-54.