Cofinite Topology

Let X is 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.


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.


It may be noted that every infinite set may or may not be co – finite topology, for this suppose X = \mathbb{R} (set of real numbers which is 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 compliments of all the members of topology along with empty set are finite.


If X is finite, then 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.