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Abstract. Under Martin's Axiom, we completely characterize the algebraic structure of Abelian groups of the size c that admit a countably compact Hausdor¤ group topology. It turns out that, in the torsion case, these are exactly the groups that admit a pseudocompact Hausdor¤ group topology, but this is no more valid for non-torsion Abelian groups. The algebraic constraints for the existence of a countably compact Hausdor¤ group topology on an Abelian group G of size c are relatively simple: for every integer n, the subgroup G½n ¼ fx A G : nx ¼ 0g of G has to be finite or satisfy jG½nj ¼ c. The same has to hold true for the subgroup dG½n, where d is any divisor of n. In addition, if G is non-torsion, then the free rank rðGÞ of G must be equal to c. 2000 Mathematics Subject Classification: 22A05, 54H11; 54A25, 54D30, 54A35. 1 Introduction Most of the theory of Abelian groups is built on the study of appropriate cardinal invariants of the groups. Similarly, the role of cardinal invariants of the topological spaces is fundamental in general topology. Therefore, it is not an exaggeration to say that cardinal invariants are the
Forum Mathematicum – de Gruyter
Published: Oct 1, 2003
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