Shape Loop Space of Pro-discrete Spaces

In this paper, considering the kth shape loop space ˇΩ p k (X, x), for an HPol∗-expansion p : (X, x) → ((Xλ, xλ), [pλλ′ ], Λ) of a pointed topological space (X, x), first we prove that ˇΩk commutes with the product under some conditions and then we show that ˇΩp k (X, x) ∼= lim ← ˇΩ p k (Xi, xi), for a pro-discrete space (X, x) = lim ← (Xi, xi) of compact polyhedra. Finally, we conclude that these spaces are metric, second countable and separable.