Results 1 to 2 of 2

Math Help - Show Compactness

  1. #1
    Junior Member
    Joined
    Sep 2009
    From
    Johannesburg, South Africa
    Posts
    71

    Show Compactness

    Let (X,d) have the property that every open cover of X has a finite subcover.
    I want to show X is compact.

    My start so far:

    If X is not compact there exist a sequence (x^{(n)})^{\infty}_{n=1} with no limit points. Then for every x \in X there exists a ball B(x,\epsilon) containing x which contains at most finitely many elements of this sequence..
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Failure's Avatar
    Joined
    Jul 2009
    From
    Zürich
    Posts
    555
    Quote Originally Posted by bram kierkels View Post
    Let (X,d) have the property that every open cover of X has a finite subcover.
    I want to show X is compact.

    My start so far:

    If X is not compact there exist a sequence (x^{(n)})^{\infty}_{n=1} with no limit points. Then for every x \in X there exists a ball B(x,\epsilon) containing x which contains at most finitely many elements of this sequence..
    Great! So this is an open cover of X; and now what? - By our assumption there exists a finite subcover, i.e. finitely many of those balls that cover X, each of which contains only finitely many elements of the infinite sequence x_n. - Isn't this a contradiction? - Methinks it is.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. show compactness
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 21st 2011, 08:24 AM
  2. compactness
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: May 19th 2010, 09:33 AM
  3. Compactness
    Posted in the Differential Geometry Forum
    Replies: 11
    Last Post: March 11th 2010, 11:17 AM
  4. Compactness; show inf is achieved
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: February 23rd 2010, 04:02 PM
  5. Topological Compactness and Compactness of a Set
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 22nd 2009, 12:16 AM

Search Tags


/mathhelpforum @mathhelpforum