Let X be an infinite set and p a point in X, chosen once and for all. Let T be the collection of subsets V of X, for which either p is not a member of V, or p is a member of V and its complement ~V is finite.

I have proved that T is a topology on X, but I don't seem to be able to prove that;

a) (X,T) is a Hausdorff space.

b) (X,T) is a compact space.