It's not clear to me why this is posted in amathematicsforum. Yes, every closed and bounded subset of a Euclidean space is compact but your argument that the universe is closed, as well as your argument that it is bounded, is physics, not mathematics. Indeed, I am not sure whether Kakutani's spaceisEuclidean but I am sure that the question of whether it is or not is one of physics, not mathematics.