Originally published at 狗和留美者不得入内. You can comment here or there.

In a complete metric space , we call a point for which is open an isolated point. If is countable and there are no isolated points, we can take , with each of the 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.