Baire space has isolated point

Question

Let $X$ be a Hausdorff, Baire space. I want to prove that $X$ has an isolated point.
In a Hausdorff space, singletons $\{x\}$ are closed.
Now suppose for a contradiction $X$ has no isolated points. Then the set $X \setminus \{x\}$ is not open. How should I proceed from here?

Answer 1

You can't prove it, since it is false. For instance, $\mathbb R$, endowed with the usual topology, is a Baire space, but it has no isolated point.