1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How to prove something is closed and bounded, ie compact

  1. Sep 25, 2012 #1
    1. The problem statement, all variables and given/known data
    I need to prove that a closed ball(radius r about x0) is closed and bounded. The same goes for a sphere(radius r about x0).


    2. Relevant equations



    3. The attempt at a solution
    How does one go about proving something is closed and bounded? My book is not very helpful and searching hasn't yielded much. This is only a part of the problem, but the rest should be doable once I get this.
     
  2. jcsd
  3. Sep 25, 2012 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Start by giving the definitions of "closed ball", "sphere", "closed" and "bounded".
     
  4. Sep 25, 2012 #3
    Well I'm lacking on the definition of "closed" and "bounded" which is what I need to get started
     
  5. Sep 25, 2012 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    What book are you using? Aren't those things in there?
     
  6. Sep 25, 2012 #5
    Ok so I assume i can prove something is bounded if every neighborhood of point along the edge(ie at r from x0) contains both a point in the ball and outside the ball. I just took that from the boundary point definition, now that doesn't imply something is closed does it?
     
  7. Sep 25, 2012 #6

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    That's not what bounded means. In fact, bounded has nothing to do with boundary. A bounded set is one that is fits inside a ball of some finite radius.

    There are various equivalent definitions of closed. You must find out which one your book is using. Some possibilities:
    * a set is closed if and only if its complement is open
    * a set is closed if and only if it contains all of its limit points
    * a set is closed if and only if it contains all of its boundary points
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: How to prove something is closed and bounded, ie compact
Loading...