Is R with the cofinite topo1o separable Prove your answer So

Is R with the cofinite topo1o separable? Prove your answer.

Solution

Theorem:
Let T=(S,?) be a finite complement topology on an infinite set S.

Then T is a separable space.

Proof:

Let S be a set.
Then S is countably infinite iff there is a bijection f:S?N, where N is the set of natural numbers.
That is, it is an infinite set of the form:
    {s0,s1,