If two topological spaces are weak homotopy equivalent to each other, are their Cech cohomology groups the same?
To give a more enlightening answer to the question:
Cech cohomology is not the same as singular cohomology. However it is on CW-complexes. But there is CW approximation for topological spaces and singular cohomology is a weak homotopy invariant, so Cech cohomology can't be.