Cofinite Topology

Let X be a non empty set, and then the collection of subsets of X whose compliments are finite along with \phi (empty set) forms a topology on X, and is called co-finite topology.

 

Example:

Let X = \left\{ {1,2,3} \right\} with topology \tau = \left\{ {\phi ,\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\},\left\{ {1,2} \right\},\left\{ {2,3} \right\},\left\{ {1,3} \right\},X} \right\} is a co–finite topology because the compliments of all the subsets of X are finite.

Note:

It may be noted that every infinite set may or may not be a co–finite topology. For this suppose X = \mathbb{R} (set of real numbers which is an infinite set) with topology \tau = \left\{ {\phi ,\mathbb{R} - \left\{ 1 \right\},\mathbb{R} - \left\{ 2 \right\},\mathbb{R} - \left\{ {1,2} \right\},\mathbb{R}} \right\} is a co–finite topology because the compliments of all the members of topology along with empty set are finite.

Remark:

If X is finite, then the topology \tau is discrete. For a subset of X belongs to \tau if and only if it is either empty or its compliment is finite. When X is finite, the compliment of each of its subset is finite and therefore, each subset of X belongs to \tau . Hence \tau is the discrete topology on X.