As I understand it, a topology T on a set X (finite or infinite) is called "the finite-closed topology" or "the cofinite topology" if the closed subsets of X are X itself, plus all the finite subsets of X; that is, the open sets are the empty set and all subsets of X which have finite complements.

I'm trying to show that T is in fact a topology. I'm having a problem.

I'm satisfied that the intersection of any two open sets in T will also be in T. So far so good. But ... what about unions?

It seems to me that many unions of sets in X would give the whole set X as its result. But X is a closed set. For instance, suppose X = { 1,2,3,4 }. One subset is A = { 1,2,3 } and another is B = { 2,3,4 }. The union of A and B would be X.

What am I missing?