Extraordinary strong homology

We develop strong shape theory for arbitrary topological pro-spaces. When restricted to spaces, our theory is equivalent to the Lisica–Mardes̆ić strong shape theory. We construct also extraordinary strong homology H∗(_ ,E) for pro-spaces with the coefficients in a spectrum. The theory is isomorphic to the Lisica–Mardes̆ić strong homology H∗(_ ,G) when E=K(G) is the Eilenberg–MacLane spectrum.