Counterexample "intersections of 2 compacts is compact"? Hello, I'm looking for a counterexample to "If A and B are compact subsets of a topological space X, then [itex]A \cap B[/itex] is compact." It's not for homework. I found one online, but it talked about "double-pointed" things which I didn't understand... My knowledge is an introductory course in topology: the first four chapters of Munkres. I realize I have to look for a non-Hausdorff space, but the only one I know --if I remember correctly--is the cofinite topology (or the related Zariski topology, but I'm not too familiar with that one).