Let X be any infinite set, given the finite complement topology (i.e. a non- empty subset A of X is open if and only if X\ A is finite). Show that X is compact. Hint for Problem 1. Let {Uafael be an open cover of X. Pick any (non-empty) Ug among these open sets. Show that, in addition to Us, you only need a finitely many Uo's to cover X.