On Topological Shape Homotopy Groups

Abstract

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 which $\check{\pi}_k^{top}$ succeeds in distinguishing the shape type of $X$ and $Y$ while $\check{\pi}_k$ fails, for all $k\in \Bbb{N}$. Moreover, we present some basic properties of topological shape homotopy groups, among them commutativity of $\check{\pi}_k^{top}$ with finite product of compact Hausdorff spaces. Finally, we consider a quotient topology on the $k$th shape group induced by the $k$th shape loop space and show that it coincides with the above topology.