Cofiniteness - Cofinite Topology

The cofinite topology (sometimes called the finite complement topology) is a topology which can be defined on every set X. It has precisely the empty set and all cofinite subsets of X as open sets. As a consequence, in the cofinite topology, the only closed subsets are finite sets, or the whole of X. Symbolically, one writes the topology as

This topology occurs naturally in the context of the Zariski topology. Since polynomials over a field K are zero on finite sets, or the whole of K, the Zariski topology on K (considered as affine line) is the cofinite topology. The same is true for any irreducible algebraic curve; it is not true, for example, for XY = 0 in the plane.