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.