  Uchillo-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 $(X,x)$,
$Sh^*((S^k,*),(X,x))=\check{\pi}_k^{*}(X,x)$, 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 that $\check{\pi}_k^{*^{top}}$ preserves finite product
of compact Hausdorff spaces. Also, we show that for a pointed topological space
$(X,x)$, $\check{\pi}_k^{top}(X,x)$ can be embedded in
$\check{\pi}_k^{*^{top}}(X,x)$.
