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.

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 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.

Remark:

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 $X$.