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
Famous quotes containing the word proof:
“If any doubt has arisen as to me, my country [Virginia] will have my political creed in the form of a “Declaration &c.” which I was lately directed to draw. This will give decisive proof that my own sentiment concurred with the vote they instructed us to give.”
—Thomas Jefferson (1743–1826)
“If we view our children as stupid, naughty, disturbed, or guilty of their misdeeds, they will learn to behold themselves as foolish, faulty, or shameful specimens of humanity. They will regard us as judges from whom they wish to hide, and they will interpret everything we say as further proof of their unworthiness. If we view them as innocent, or at least merely ignorant, they will gain understanding from their experiences, and they will continue to regard us as wise partners.”
—Polly Berrien Berends (20th century)
“Right and proof are two crutches for everything bent and crooked that limps along.”
—Franz Grillparzer (1791–1872)