D Dikranjan, D Shakhmatov
MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY 133 633 VIII - 83 1998年05月
[査読有り] 研究論文（学術雑誌）
Looking for a natural generalization of compact spaces, in 1948 Hewitt introduced pseudocompact spaces as those Tychonoff spaces on which every real-valued continuous function is bounded. The algebraic structure of compact Abelian groups was completely described in the fifties and sixties by Kaplansky, Harrison and Hulanicki. In this paper we study systematically the algebraic structure of pseudocompact groups, or equivalently, the following problem: Which groups can be equipped with a pseudocompact topology turning them into topological groups? We solved this problem completely for the following classes of groups: free groups and free Abelian groups (or more generally, free groups in some variety of abstract groups), torsionfree Abelian groups (or even Abelian groups G with \G\ = r(G)), torsion Abelian groups, and divisible Abelian groups.
Even though out main problem deals with the existence of some topologies on groups, it has a strong set-theoretic flavor. Indeed, the existence of an infinite pseudocompact group of cardinality tau and weight sigma is equivalent to the following purely set-theoretic condition Ps(tau, sigma) introduced by Cater, Erdos and Galvin for entirely different purposes: The set {0, 1)(sigma) of all functions from (a set of cardinality) a to the two-point set {0, 1} contains a subset of size tau whose projection on every countable subproduct {0, 1}(A) is a surjection. Despite its innocent look, the problem of which cardinals sigma and tau enjoy such a relationship is far from being solved, and is closely related to the Singular Cardinal Hypothesis.
A variety of necessary conditions, both of algebraic and of set-theoretic nature, for the existence of a pseudocompact group topology on a group is discovered. For example, pseudocompact torsion groups are locally finite. If an infinite Abelian group G admits a pseudocompact group topology of weight sigma, then either r(p)(r(G),a) or Ps(rp(G),a) for some prime number p must hold, where r(G) and rp(G) are the free rank and the prank of G respectively. If an Abelian group G has a pseudocompact group topology, then \{ng : g is an element of G}\ less than or equal to 2(2r(G)) for some n. This yields the inequality \G\ less than or equal to 2(2r(G)) for a divisible pseudocompact group.
Turning to necessary and sufficient conditions, we show that a nontrivial Abelian group G admits a connected pseudocompact group topology of weight a if and only if \G\ less than or equal to 2(sigma) and Ps(r(G),sigma) hold. Moreover, a free group with tau generators in a variety nu of groups admits a pseudocompact group topology if and only if Ps(tau, sigma) holds for some infinite sigma, and the variety nu is generated by its finite groups. It should be noted that most of the classical varieties of groups have the last property, the only exception the authors are aware of being the Burnside varieties B-n for odd n > 665.