proof of Baire category theorem

Let (X,d) be a complete metric space, and Uk a countableMathworldPlanetmath collectionMathworldPlanetmath of dense, open subsets. Let x0X and ϵ0>0 be given. We must show that there exists a xkUk such that


Since U1 is dense and open, we may choose an ϵ1>0 and an x1U1 such that


and such that the open ballPlanetmathPlanetmath of radius ϵ1 about x1 lies entirely in U1. We then choose an ϵ2>0 and a x2U2 such that


and such that the open ball of radius ϵ2 about x2 lies entirely in U2. We continue by inductionMathworldPlanetmath, and construct a sequence of points xkUk and positive ϵk such that


and such that the open ball of radius ϵk lies entirely in Uk.

By construction, for 0j<k we have


Hence the sequence xk,k=1,2, is Cauchy, and convergesPlanetmathPlanetmath by hypothesisMathworldPlanetmath to some xX. It is clear that for every k we have


Moreover it follows that


and hence a fortiori


for every k. By construction then, xUk for all k=1,2,, as well. QED

Title proof of Baire category theorem
Canonical name ProofOfBaireCategoryTheorem
Date of creation 2013-03-22 13:06:55
Last modified on 2013-03-22 13:06:55
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 10
Author rmilson (146)
Entry type Proof
Classification msc 54E52