Proof
The following is a standard proof that a complete pseudometric space is a Baire space.
Let be a countable collection of open dense subsets. We want to show that the intersection is dense. A subset is dense if and only if every nonempty open subset intersects it. Thus, to show that the intersection is dense, it is sufficient to show that any nonempty open set has a point in common with all of the . Since is dense, intersects ; thus, there is a point and such that:
where and denote an open and closed ball, respectively, centered at with radius . Since each is dense, we can continue recursively to find a pair of sequences and such that:
(This step relies on the axiom of choice.) Since when, we have that is Cauchy, and hence converges to some limit by completeness. For any, by closedness,
Therefore and for all .
Read more about this topic: Baire Category Theorem
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