Shape Loop Space of Pro-discrete Spaces

In this paper, considering the kth shape loop space \check{\Omega}_{k}^{\mathbf{p}}(X,x), for an HPol_*-expansion \mathbf{p}:(X,x)\rightarrow ((X_{\lambda},x_{\lambda}),[p_{\lambda\lambda'}],\Lambda) of a pointed topological space (X,x), first we prove that  \check{\Omega}_{k} commutes with the product under some conditions and then  we show that \check{\Omega}_k^{\mathbf{p}}(X,x)\cong \displaystyle{\lim_{\leftarrow}\check{\Omega}_k^{\mathbf{p}}(X_i,x_i)}, for a pro-discrete space (X,x)=\displaystyle{\lim_{\leftarrow}(X_i,x_i)} of compact polyhedra. Finally,  we conclude that these spaces are metric, second countable and separable.

Keywords:

Shape theory , Inverse limit , Loop space