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I already proved that this satisfies the conditions for defining a topology (called the finite complement topology) But I am having a lot of trouble with defining when a point is a limit. I know that any open subset (defined by tau) containing such a limit point must have a non-empty intersection with A for it to be a limit point.

I have tried considering multiple cases: I know if X is finite then it is the discrete topology and therefore no limit points exist. So the non trivial case if when X is infinite.

Now, if A is finite then for any open subset containing x, O, the complement with A would obviously be finite so therefore non empty?

I'm sort of stuck and could use a nudge in some helpful direction.