Baire Category Theorem

Theorem
The Baire Category Theorem states that every complete metric space is a Baire space.

Proof
Let $$U_n$$ be a countable collection of open dense subsets.

We want to show that the intersection $$\bigcap U_n$$ is dense.

A subset $$S$$ is dense if every nonempty open subset intersects it. To see this, take any point $$x$$ in the space, and consider an open neighbourhood of this point. This neighbourhood contains an open set $$V$$ such that $$x\in V$$, which intersects $$S$$ as it is open and nonempty. So every neighbourhood of every point intersects $$S$$, and so $$S$$ is dense.

Thus, to show that the intersection is dense, it is sufficient to show that any nonempty open set $$W$$ has a point $$x$$ in common with all of the $$U_n$$.

Since $$U_1$$ is dense, $$W$$ intersects $$U_1$$; thus, there is a point $$x_1$$ and $$r_1 > 0$$ such that:
 * $$\overline{B}(x_1, r_1) \subset W \cap U_1$$.

where $$B(x, r)$$ and $$\overline{B}(x, r)$$ denote an open ball centered at $$x$$ with radius $$r$$ and its closure, respectively.

Since $$U_n$$ are dense, in a recursive manner, we find a pair of sequences $$x_n$$ and $$r_n > 0$$ such that:
 * $$\overline{B}(x_n, r_n) \subset B(x_{n-1}, r_{n-1}) \cap U_n$$

as well as $$r_n < 1/n $$.

Since $$x_n \in B(x_m, r_m)$$ when $$n > m$$, we have that $$x_n$$ is a Cauchy Sequence, and $$x_n$$ converges to some limit $$x$$ by completeness.

For any $$n$$, by closedness:
 * $$x \in \overline{B}(x_{n+1}, r_{n+1}) \subset B(x_n, r_n)$$.

Hence, $$x \in W$$ and $$x \in U_n$$ for all $$n$$.