A nice consequence of Baire category theorem

In a complete metric space X, we call a point x for which \{x\} is open an isolated point. If X is countable and there are no isolated points, we can take \displaystyle\cap_{x \in X} X \setminus x = \emptyset, with each of the X \setminus x open and dense, to violate the Baire category theorem. From that, we can arrive at the proposition that in a complete metric space, no isolated points implies that the space uncountable, and similarly, that countable implies there is an isolated point.