# 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 $$X$$.