Cofinite topology (finite complement topology)

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}

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.