On countably compact topologies on compact groups and on dyadic compacta

Abstract

In this paper we apply certain classical results of S. Mazur, J. Keisler, A. Tarski and N.Th. Varopoulos to the theory of countably compact topological groups. For example, under a mild restriction on the cardinality of a compact group G, it is proved that there is no strictly stronger countably compact group topology on G. If f is a one-to-one continuous mapping of a countably compact topological group of countable tightness onto a compact Hausdorff space X, then X is metrizable. If a countably compact topological group of countable tightness acts continuously and transitively on a compact Hausdorff space X, then X is metrizable. Among the questions left unsettled is this: Let f be a sequentially continuous homomorphism of a compact topological group G onto a topological group H. Is then H countably compact? Equivalently, is then H totally bounded? Under Martin's axiom this question is answered positively.