Lebegue Covering Lemma - covers X entirely So let δ = min...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
The Lebesgue Covering Lemma copied from http://mathblather.blogspot.com/ February 3, 2012 Theorem(Lebesgue Covering Lemma): Given that X is a compact metric space, let U be a open covering of X . Then there exist δ > 0 such that for all x X , the ball B δ ( x ) U for some U ⊂ U . Proof. Take any x X , then x U x for some U x ⊂ U , since U x is open, there is a ball B ( x ) ( x ) U x . Now the set of all balls of the form: { B ( x ) / 2 ( x ) : x X } is an open cover of X , so by compactness of X , there is a finite collection of balls: B ( x 1 ) / 2 ( x 1 ) , . . . , B ( x n ) / 2 ( x n ), where x 1 , . . . , x
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: covers X entirely. So let δ = min { ± ( x 1 ) / 2 ,...± ( x n ) / 2 } . Pick any x ∈ X , I then claim that B δ ( x ) ⊂ U for some U ⊂ U . Since x ∈ B ± ( x j ) / 2 ( x j ) for some x j , we take any y ∈ B δ ( x ), by triangular inequality: d ( y,x j ) ≤ d ( y,x ) + d ( x,x j ) < δ + d ( x,x j ) ≤ ± ( x j ). Hence y ∈ B ± ( x j ) ( x j ), this implies that B δ ( x ) ⊂ B ± ( x j ) ( x j ) ⊂ U x j 1...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern