Topological coarse shape homotopy groups

Abstract

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⁎((Sk,⁎),(X,x))=πˇk⁎(X,x), which makes it a Hausdorff topological group. Moreover, we study some properties of these topological coarse shape homotopy groups such as second countability, movability and in particular, we prove that πˇk⁎top preserves finite product of compact Hausdorff spaces. Also, we show that for a pointed topological space (X,x), πˇktop(X,x) can be embedded in πˇk⁎top(X,x).