Cofinite topology

Definition
Suppose $$X$$ is a set. The cofinite topology on $$X$$ is a topological space structure on $$X$$ that can be defined in the following equivalent ways:

If the set is finite, the cofinite topology makes it a discrete space.

Facts

 * A set equipped with the cofinite topology is a Toronto space.
 * Any bijection between two sets is a homeomorphism between them as topological spaces with the cofinite topology.
 * The cofinite topology is the Zariski topology on any connected one-dimensional algebraic variety over a field.
 * Any set equipped with the cofinite topology is a Noetherian space, and hence a hereditarily compact space.