Prove that the digital line in not homeomorphic to Z with the finite complement topology
A finite complement topology on is compact. Any open cover of has a finite subcover. Once you have chosen an arbitrary initial open set of a finite complement topology on , only finite point sets are remained to cover the whole . The remaining choices of open sets for an open cover for is bound to be finite.
In contrast, the digital topology on is not compact. All basis elements of a digital topology on forms an open cover for , but it has no finite subcover.
Thus, the above two topological spaces are not homeomorphic.