Cancellation of topological groups

摘要

Let m be a family of Hausdorff topological groups. A Hausdorff topological group P is called m-cancellable if for any two topological groups G and H in m, the product group G x P is topologically isomorphic to H x P if and only if G and H are topologically isomorphic. If m is the class of all Hausdorff topological groups, m-cancellable topological groups are called cancellable topological groups.We show in this paper that every topological group in a family n of topological groups is cancellable. In particular, the additive group R of reals and Q of rationals endowed with the usual topologies are both cancellable. We also show that every topological group in a family m of topological groups is cancellable. In particular, every topologically simple non-abelian group is cancellable. It is also proved that every topologically simple Hopfian or co-Hopfian topological group is cancellable. (C) 2019 Elsevier B.V. All rights reserved.