Share
Explore BrainMass

Real Analysis : Baire Category Theorem

If {G1,G2,G3,...} is a countable collection of dense, open sets then the intersection (U top infinity bottom n=1)G_n is not empty.

Solution Preview

real analysis
Posting

if {G1,G2,G3,...} is a countable collection of dense, open sets then the intersection(U top infinity bottom n=1)G_n is not empty.
-------------------------------------
I am using "à?" for the Intresection Symbol "U"
I will also use o1,o2,o3.,,,, instead of G's..

Baire Category Theorem
----------------------
Let s be a complete metric space, and let o1 o2 o3 etc be a countable collection of dense open sets in s. Let o be the intersection of all these open sets. We will show that o is nonempty and dense in s. This is the Baire category theorem.

Let w be an arbitrary nonempty open set in s, and intersect w with o1, having x1 in the intersection. Let s1 be an open ball in the intersection, with radius r1 centered at x1. Remember that the open ball s1 is the set of points less than r1 distance from x1. Let c1 be the closure of s1, and remember that c1 cannot include any points farther than r1 from x1.

Cut r1 in half, if need be, so that c1 lies entirely inside the intersection of o1 and w.

Since o2 is dense, o2 intersects s1. As before, the intersection is a nonempty open set. Let x2 lie in the intersection, and build an open ball s2 about x2 with radius r2, subject to the following criteria.

Shrink r2 so that it is less than half of r1. We want the diameters to approach 0.

Let d2 be the distance from x1 to x2. We know d2 < r1. Shrink r2 so that it is less than r1-d2. This assures us that c2, the ...

Solution Summary

A proof using the Baire Category Theorem is provided. The solution is detailed.

$2.19