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 be a metric space with complete metric . Suppose that is not a Baire space. Then there is a countable collection of dense open subsets of such that the intersection is not dense in .
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 if .
We construct a sequence of open balls satisfying
as follows: By hypothesis, the interior of is not empty (otherwise would be dense in ), so we may choose an open ball with such that . Given , the set is nonempty, because is dense in , and it is open, because and are open. This allows us to choose an open ball as desired.
Note that by construction for , thus . Therefore, the sequence is Cauchy and has a limit point by completeness. In the limit , we obtain (strictness is lost), hence for all . This shows that for all , that is . On the other hand, , in contradiction to the preceding statement.
Baire Category Theorem #2
Every locally compact Hausdorff space is a Baire space.