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:

Type of description	Details
in terms of open subsets	The open subsets are precisely the empty set and the cofinite subsets, i.e., the subsets whose complements are finite subsets of $X$.
in terms of closed subsets	The closed subsets are precisely the whole space and the finite subsets.
in terms of a subbasis	We can take as a subbasis the complements of singleton subsets in the space.
as a coarsest topology	The coarsest topology on $X$ for which it is a T1 space.

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.