Since the only algebraic subsets of a one-dimensional affine space (which are not the space itself or the empty set) are finite subsets, the Zariski and cofinite topologies coincide for these spaces.
The Zariski and cofinite topologies also happen to be two of the best examples for topologies which are most simply defined in terms of their closed sets, rather than their open sets. The key observation which is the source of my question is that their being defined most simply in terms of closed sets also seems to correspond to the fact that they coincide in special cases.
Is this a recurring theme? Do topologies which are most simply defined in terms of closed sets tend to coincide in special cases? Do they have or tend to have any distinctive properties (besides the aforementioned one of being most simply defined by their closed sets) which separate or tend to separate them from other topologies (which are best defined in terms of open sets or bases or sub-bases or neighborhood systems)? Or am I simply inferring too much from a single example?