Isolated points in a complete metric space

Question

I need to prove that if $X$ is a countable (infinite) complete metric space, then $X$ has infinitely many isolated points.

I have read that the Baire Category theorem implies that $X$ should have at least one isolated point, but I have no idea how to show the set of isolated points is infinite.

Thanks for any help!

Answer 1

Suppose that it has a finite number of isolated points $x_1,..,x_n$, $Y=X-\{x_1,..,x_n\}$ is still complete. Every element $y\in Y$ as an empty interior. You can write $Y=\cup_{n\in\mathbb{N}}\{y_n\}$, Baire implies that $\cup_{n\in\mathbb{N}}\{y_n\}=Y$ has an empty interior. Contradiction.