The reason complete metric spaces are so important in functional analysis.
- Baire's category theorem:
- In a complete metric space, the intersection of countably many dense open subsets is dense.
In particular, (in a non-empty space...) the intersection is non-empty. This statement has the full force of the original statement, since we may limit ourselves to any open set, and find that the intersection includes a point from that open set.
Often the quantifiers in the statement are reversed: call any countable union of nowhere dense sets a set of first category, and all other sets second category. Then Baire's theorem states that a complete metric space is of second category.