An infinite set with the cofinite topology is not Hausdorff.

Question

I try to demostrate that an infinite set $X$ with the cofinite topology is not Hausdorff. I know $A⊂X$ can be written as the intersection of open sets containing it. But I don´t know how to get a contradiction.

Answer 1

The complement of an open set is finite. Can another open set fit in the complement?

Answer 2

Hint: The intersection of any two non-empty open contains all but finitely many points.