#### definition

Let X be a non-empty set and τ be the collection of φ and all subsets of X whose complements are finite. Then τ is called cofinite topology on X (also called finite complement topology) and the pair (X, τ) is called cofinite topological space.

τ_{cof }= {φ ⋀ U⊂X | U^{c }is finite}

**OR**

τ_{cof}= {U⊂X | U^{c }is finite or X}

**Note**

If set X is finite then the co-finite topology on set X is the discrete topology.

**Co-finite topology** **Proof**

**Show that τ**_{cof }= {φ⋀ U⊂X | U^{c }is finite}

_{cof }= {φ⋀ U⊂X | U

^{c }is finite}

**Proof**

This shows that τ is a topology on set X. this is called cofinite topology and the pair (X, τ_{cof}) is called the cofinite topological space.

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