The Eckman–Hilton duality [4] reverses arrows in diagrams, turns products to co-products, and multiplications to co-multiplications, etc. In accordance with this process, Kan [5] obtained the dual of a monoid structure in the category of groups. In this way, we obtain co-monoid structures on topological groups. The main result of this paper is that for kaω groups (see §2), we obtain a one-to-one correspondence between the co-monoid structures, and the free topological bases of the group (§3), thus obtaining topological analogues of the main results of [5].