Baire Spaces

Definition (Baire Space)

A topological space is said to be a Baire space, if any of the following equivalent conditions holds:

The intersection of countably many dense open subsets is still dense.

The union of countably many closed subsets with empty interior has empty interior.

Note that a set is dense in a topological space if and only if its complement has empty interior.

Any sufficient condition for a topological space to be a Baire space constitutes a Baire Category Theorem, of which there are several. Here we give one that is commonly used in functional analysis.

Baire Category Theorem #1

Every complete metric space is a Baire space.

Proof Let $X$ be a metric space with complete metric $d$ . Suppose that $X$ is not a Baire space. Then there is a countable collection ${U_{n}}$ of dense open subsets of $X$ such that the intersection $U := ⋂ U_{n}$ is not dense in $X$ .

In a metric space, any nonempty open set contains an open ball. It is also true, that any nonempty open set contains a closed ball, since $\overline{B (y, δ_{1})} \subset B (y, δ_{2})$ if $δ_{1} < δ_{2}$ .

We construct a sequence $(B_{n})$ of open balls $B_{n} := B (x_{n}, ϵ_{n})$ satisfying

\overline{B_{n + 1}} \subset B_{n} \cap U_{n} ϵ_{n} < \frac{1}{n} \forall n \in N,

as follows: By hypothesis, the interior of $X ∖ U$ is not empty (otherwise $U$ would be dense in $X$ ), so we may choose an open ball $B_{1}$ with $ϵ_{1} < 1$ such that $\overline{B_{1}} \subset X ∖ U$ . Given $B_{n}$ , the set $B_{n} \cap U_{n}$ is nonempty, because $U_{n}$ is dense in $X$ , and it is open, because $B_{n}$ and $U_{n}$ are open. This allows us to choose an open ball $B_{n + 1}$ as desired.

Note that by construction $B_{m} \subset B_{n}$ for $m \geq n$ , thus $d (x_{m}, x_{n}) < ϵ_{n} < \frac{1}{n}$ . Therefore, the sequence $(x_{n})$ is Cauchy and has a limit point $x$ by completeness. In the limit $m \to \infty$ , we obtain $d (x, x_{n}) \leq ϵ_{n}$ (strictness is lost), hence $x \in \overline{B_{n}}$ for all $n$ . This shows that $x \in U_{n}$ for all $n$ , that is $x \in U$ . On the other hand, $x \in \overline{B_{1}} \subset X ∖ U$ , in contradiction to the preceding statement. $□$

Baire Category Theorem #2

Every locally compact Hausdorff space is a Baire space.