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?
Check the definition of a topology. The topology must contain the full set and the empty set. Since the compliment of the full set is the empty set and the empty set is the compliment of the full set, both sets are simultaneously open and closed. They are clopen sets.
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