On topological shape homotopy groups

In this paper, using the topology on the set of shape morphisms between arbitrary topological spaces X, Y  , Sh(X,Y)Sh(X,Y), defined by Cuchillo-Ibanez et al. in 1999, we consider a topology on the shape homotopy groups of arbitrary topological spaces, which make them Hausdorff topological groups. We then exhibit an example in which πˇktop succeeds in distinguishing the shape type of X and Y   while πˇk fails, for all k∈Nk∈N. Moreover, we present some basic properties of topological shape homotopy groups, among them commutativity of πˇktop with finite product of compact Hausdorff spaces. Finally, we consider a quotient topology on the kth shape group, induced by the kth shape loop space and show that it coincides with the above topology.