# 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 $B(y,δ_{1}) ⊂B(y,δ_{2})$ if $δ_{1}<δ_{2}$.

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

$B_{n+1} ⊂B_{n}∩U_{n}ϵ_{n}<n1 ∀n∈N,$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 $B_{1} ⊂X∖U$. Given $B_{n}$, the set $B_{n}∩U_{n}$ 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 $B_{m}⊂B_{n}$ for $m≥n$, thus $d(x_{m},x_{n})<ϵ_{n}<n1 $. Therefore, the sequence $(x_{n})$ is Cauchy and has a limit point $x$ by completeness. In the limit $m→∞$, we obtain $d(x,x_{n})≤ϵ_{n}$ (strictness is lost), hence $x∈B_{n} $ for all $n$. This shows that $x∈U_{n}$ for all $n$, that is $x∈U$. On the other hand, $x∈B_{1} ⊂X∖U$, in contradiction to the preceding statement. $□$

Baire Category Theorem #2

Every locally compact Hausdorff space is a Baire space.