Counting compact group topologies

摘要

Given a group K, the symbol cgt(K) denotes the set of Hansdorif compact group topologies on K. The authors ask: when K = k >= w, what are the possible cardinalities of a pairwise homeomorphic subset [resp., a pairwise nonhomeomorphic subset] of cgt(K)? Revisiting (sometimes, improving) theorems of Halmos, Hulanicki, Fuchs, Hawley, Chuan/Liu, Kirku, and Shtern, the authors show inter alia: