Prove that the digital line in not homeomorphic to Z with the finite complement topology
If two topological spaces are homeomorphic to each other, they share the same topological invariants, such as connectedness, compactness, Hausdorffness, and homotopy groups. That means if one topological space is compact but the other topological space is not compact, then two topological spaces are not homeomorphic to each other.
A finite complement topology on $\displaystyle \mathbb{Z}$ is compact. Any open cover of $\displaystyle \mathbb{Z}$ has a finite subcover. Once you have chosen an arbitrary initial open set of a finite complement topology on $\displaystyle \mathbb{Z}$, only finite point sets are remained to cover the whole $\displaystyle \mathbb{Z}$. The remaining choices of open sets for an open cover for $\displaystyle \mathbb{Z}$ is bound to be finite.
In contrast, the digital topology on $\displaystyle \mathbb{Z}$ is not compact. All basis elements of a digital topology on $\displaystyle \mathbb{Z}$ forms an open cover for $\displaystyle \mathbb{Z}$, but it has no finite subcover.
Thus, the above two topological spaces are not homeomorphic.