Finite Compliment Topology
This should probably be an easy one.
The "finite compliment topology" is defined by my text as the collection of sets U of a set X such that X - U is either finite or all of X.
Nothing in the definition says that X must be finite, but I was looking at an example of a finite set and it struck me that the FCT on a finite set X is the same as the discrete topology on that same X. (The discrete topology is defined as the collection of all subsets of X.) I can verify this for a given set X but I am at a loss of how to prove it in general. I've been toying with some kind of construction or an induction proof, but can't quite seem to get a good start on it. Any thoughts?