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:
“There is no better proof of a man’s being truly good than his desiring to be constantly under the observation of good men.”
—François, Duc De La Rochefoucauld (1613–1680)
“The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.”
—Andrew Michael Ramsay (1686–1743)
“The thing with Catholicism, the same as all religions, is that it teaches what should be, which seems rather incorrect. This is “what should be.” Now, if you’re taught to live up to a “what should be” that never existed—only an occult superstition, no proof of this “should be”Mthen you can sit on a jury and indict easily, you can cast the first stone, you can burn Adolf Eichmann, like that!”
—Lenny Bruce (1925–1966)