Algebraic structure of the countably compact groups
by
Dikran Dikranjan
Department of Mathematics and Computer Science, Udine University
Coauthors: Michael Tkachenko (Departamento de Matemáticas, Universidad Autónoma Metropolitana, Mexico City, Mexico)

The problem of characterization of the class of abelian groups that admit compact group topologies was set by Halmos in 1944 and resolved in the next decade by the common efforts of Kaplansky, Harison and Hulanicki. The counterpart of Halmos' problem for pseudocompact groups was faced by various authors ([1-5]) after the pioneering work of van Douwen [6] where the first restraint was given on the cardinality of an infinite pseudocompact group. The present talk will treat the counterpart of Halmos' problem for small countably compact groups. More precisely, we develop the construction from [7] to give a complete description, under the assumption of MA, of the class of all abelian groups of size at most continuum that admit countably compact group topologies. For torsion groups, the description coincides with that for the case of pseudocompact group topologies given in [DS1, DS2].

References

[1] W. Comfort and J. van Mill, Concerning connected, pseudocompact Abelian groups, Topology Appl. 33 (1989), 21-45.

[2] W. Comfort and D. Remus, Imposing pseudocompact group topologies on Abelian groups, Fund. Math. 142 (1993), 221-240.

[3] W. Comfort and D. Remus, Abelian torsion groups with a pseudocompact group topology, Forum Math. 6 (1994), 323-337.

[4] D. Dikranjan and D. Shakhmatov, Pseudocompact topologizations of groups, Zb. radova Filozof. fakulteta u Nisu, Ser. Mat. 4 (1990), 83-93.

[5] D. Dikranjan and D. Shakhmatov, Algebraic structure of the pseudocompact groups, Memoirs Amer. Math. Soc., 133/633, April 1998.

[6] E. K. van Douwen, The weight of a pseudocompact (homogeneous) space whose cardinality has countable cofinality, Proc. Amer. Math. Soc. 80 (1980), 678-682.

[7] M. Tkachenko, On countably compact and pseudocompact topologies on free Abelian groups, Soviet Math. (Izv. VUZ) 34 (5) (1990), 79-86.

Algebraic structure of small countably compact Abelian groups

Date received: June 13, 2000

Copyright © 2000 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 # caeh-21.