Hartshorne's Connectedness Theorem asserts that, if a local ring (R,m) has depth at least 2, then Spec(R)\{m} is connected. On the other hand, Faltings' Connectedness Theorem says, if an ideal I of a d-dimensional complete local domain (R,m) can be generated by d-2 elements, then Spec(R/I)\{m} is connected.