Results 1 to 9 of 9

Math Help - prove that E=union of E_i is compact

  1. #1
    Member
    Joined
    Feb 2010
    Posts
    146

    prove that E=union of E_i is compact

    If E_1..., E_n are compact, prove that E= \cup i=1 to n E_i is 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 tn11631 View Post
    If E_1..., E_n are compact, prove that E= \cup i=1 to n E_i is compact.
    I mean, if you know the Heine-Borel theorem this is easy as pi.

    The finite union of closed sets is closed and

    \text{diam }E_1\cup\cdots\cup E_n\leqslant\sum_{j=1}^{n}\text{diam }E_j<\infty.

    Ta-da!
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Feb 2010
    Posts
    146
    Quote Originally Posted by Drexel28 View Post
    I mean, if you know the Heine-Borel theorem this is easy as pi.

    The finite union of closed sets is closed and

    \text{diam }E_1\cup\cdots\cup E_n\leqslant\sum_{j=1}^{n}\text{diam }E_j<\infty.

    Ta-da!
    Ah yes! lol I feel so inferior right now for not thinking of these things! lol I'm clearly getting overtired. Thanks again your such a big help!
    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 tn11631 View Post
    Ah yes! lol I feel so inferior right now for not thinking of these things! lol I'm clearly getting overtired. Thanks again your such a big help!
    Don't feel inferior! We all have bad days! :P
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member Tinyboss's Avatar
    Joined
    Jul 2008
    Posts
    433
    It's pretty quick from the finite-open-cover definition of compactness, too; any open cover of the union is also an open cover for each compact set, and finite*n=finite.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,444
    Thanks
    1863
    Are you given that these are subsets of the real numbers (with the usual topology)? If not then you cannot use "Heine-Borel".
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Feb 2010
    Posts
    146
    Quote Originally Posted by HallsofIvy View Post
    Are you given that these are subsets of the real numbers (with the usual topology)? If not then you cannot use "Heine-Borel".
    Um the original problem posted was the only information given.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by HallsofIvy View Post
    Are you given that these are subsets of the real numbers (with the usual topology)? If not then you cannot use "Heine-Borel".
    I don't think the OP is meaning to be that advanced. I think this is a beginning analysis course.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,966
    Thanks
    1785
    Awards
    1
    Quote Originally Posted by tn11631 View Post
    If E_1..., E_n are compact, prove that E= \cup i=1 to n E_i is compact.
    Give a straightforward proof.
    Suppose that the collection  \left\{O_{\alpha}\right\} is an open cover for \bigcup\limits_{j = 1}^n {E_j } .
    For each m,~1\le m\le n we have E_m\subseteq\bigcup\limits_{j = 1}^n {E_j }
    So E_m is covered by a finite subcollection of  \left\{O_{\alpha}\right\}.
    Now the finite union of finite collections is a finite collection.
    That proves that \bigcup\limits_{j = 1}^n {E_j } is compact.

    BTW: This proof works in general topological spaces.
    Last edited by Plato; March 29th 2010 at 03:43 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. How to prove O is compact in M?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: June 3rd 2011, 07:50 AM
  2. Finite union of compact sets is compact
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 8th 2011, 08:43 PM
  3. how to prove that x: 0<=f(x)<=1 is compact
    Posted in the Differential Geometry Forum
    Replies: 11
    Last Post: April 6th 2010, 05:39 PM
  4. Compact spaces - Union and Intersection
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: March 9th 2010, 02:19 PM
  5. Replies: 2
    Last Post: April 6th 2007, 06:48 PM

Search Tags


/mathhelpforum @mathhelpforum