Countable compactness and finite powers of topological groups without convergent sequences

Abstract

We show under MA countable that for every positive integer n there exists a topological group G without non-trivial convergent sequences such that G n is countably compact but G n+1 is not. This answers the finite case of Comfort's Question 477 in the Open Problems in Topology. We also show under MA countable +2 < c = c that there are 2 c non-homeomorphic group topologies as above if n⩾2. We apply the construction on suitable sets, answering the finite case of a question of D. Dikranjan on the productivity of suitability and in a topological game defined by Bouziad.