Results 1 to 6 of 6

Math Help - Intersection of compact sets

  1. #1
    MHF Contributor harish21's Avatar
    Joined
    Feb 2010
    From
    Dirty South
    Posts
    1,036
    Thanks
    10

    Intersection of compact sets

    Given, two compact sets A and B, how do we show that their intersection A \bigcapB is also compact?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by harish21 View Post
    Given, two compact sets A and B, how do we show that their intersection A \bigcapB is also compact?
    In what kind of space?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor harish21's Avatar
    Joined
    Feb 2010
    From
    Dirty South
    Posts
    1,036
    Thanks
    10
    Quote Originally Posted by Drexel28 View Post
    In what kind of space?

    A and B are compact subsets of R
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by harish21 View Post
    A and B are compact subsets of R
    Think closed subspace
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor harish21's Avatar
    Joined
    Feb 2010
    From
    Dirty South
    Posts
    1,036
    Thanks
    10
    Quote Originally Posted by Drexel28 View Post
    Think closed subspace
    A and B are closed and bounded because they are compact.
    So A \bigcap B is also closed.
    How do we show boundedness?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by harish21 View Post
    A and B are closed and bounded because they are compact.
    So A \bigcap B is also closed.
    How do we show boundedness?
    The Heine-Borel theorem is kind of overkill here...but.

    Clearly if C\subseteq D then \text{diam }C\leqslant \text{diam }D, right? A\cap B\subseteq A
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Finite union of compact sets is compact
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 8th 2011, 08:43 PM
  2. Compact spaces - Union and Intersection
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: March 9th 2010, 02:19 PM
  3. the intersection of a collection of compact sets is compact
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: February 28th 2010, 02:58 PM
  4. intersection of compact sets
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: October 27th 2009, 05:40 PM
  5. Compact Sets
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 22nd 2008, 09:48 PM

Search Tags


/mathhelpforum @mathhelpforum