In this paper, using the topology on the set of shape morphisms between arbitrary topological spaces 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 whichπ top k succeeds in distinguishing the shape type of X and Y whileπ k fails, for all k ∈ N. Moreover, we present some basic properties of topological shape homotopy groups, among them commutativity ofπ top k 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.

The paper is devoted to the study of coarse shape of Cartesian products of topological spaces. If the Cartesian product of two spaces X and Y admits an HPol-expansion, which is the Cartesian product of HPol-expansions of these spaces, then X × Y is a product in the coarse shape category. As a consequence, the Cartesian product of two compact Hausdorff spaces is a product in the coarse shape category. Finally, we show that the shape groups and the coarse shape groups commute with products under some conditions.2000 Mathematics Subject Classification. 55P55; 18A30.

Cuchillo-Ibanez et al. introduced a topology on the sets of shape morphisms between arbitrary topological spaces in 1999. In this paper, applying a similar idea, we introduce a topology on the set of coarse shape morphisms Sh * (X, Y ), for arbitrary topological spaces X and Y . In particular, we can consider a topology on the coarse shape homotopy group of a topological space (X, x), Sh, which makes it a Hausdorff topological group. Moreover, we study some properties of these topological coarse shape homotopoy groups such as second countability, movability and in particullar, we prove thatπ * top k preserves finite product of compact Hausdorff spaces. Also, we show that for a pointed topological space (X, x), π top k (X, x) can be embedded inπ * top k (X, x).

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