Results 1 to 8 of 8

Math Help - Understanding Topology Proof

  1. #1
    Senior Member I-Think's Avatar
    Joined
    Apr 2009
    Posts
    288

    Understanding Topology Proof

    Studying on my own currently and having trouble understanding a topology proof.

    Theorem: Compact subsets of metric spaces are closed

    Proof
    Let K be a compact subset of a metric space X. We shall prove that the complement of K is an open subset of X
    Suppose p \in{X}, p \notin K. If q\in{K}, let V_q and W_q be neighbourhoods of p and q respectively, of radius less than \frac{1}{2}d(p,q)
    Since K is compact, there are finitely many points q_1,...,q_n in K such that

    K\subset{W_{q_1}\cup...\cup{W_{q_n}}=W

    If V=V_{q_1}\cap...\cap{V_{q_n}}, then V is a neighbourhood of p which does not intersect W. Hence V\subset{K^c}, so that p is an interior point of K^c. The theorem follows.

    Question
    I am confused as to how V_q and W_q are defined
    Is it that
    W_{q_i}:[\forall{p}: d(p,q_i)<\frac{d(p,q)}{2}]
    V_{q_i}:[\forall{q}: d(p,q_i)<\frac{d(p,q)}{2}]

    Also, how does the chosen radius play a role in the proof?
    Last edited by I-Think; July 18th 2011 at 05:53 PM. Reason: Fixing typos
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Feb 2008
    Posts
    410

    Re: Understanding Topology Proof

    The idea is that for each point q\in K, there are disjoint (i.e. non-overlapping) neighborhoods V_q and W_q of p and q, respectively. (Notice that p is fixed but q can be any point in K.) In order to ensure they are disjoint, we need to make sure they have a small enough radius. How small? Well if they are no more than half the distance between p and q, that will be small enough. So it doesn't really matter exactly how V_q,W_q are defined as long as they are disjoint neighborhoods. If you like we can define them as you have done. But that's just one possible choice. All we need them to be are disjoint neighborhoods.

    And then of course we have \mathcal{C}=\{W_q:q\in K\} an open cover of K. By compactness there is a finite subcover \mathcal{C}^* which is a finite set of the form \{W_{q_1},\cdots,W_{q_n}\} for some q_1,\cdots,q_n\in K. Remember that each W_{q_i} is disjoint from each V_{q_i}. So V:=\bigcap_{i=1}^n V_{q_i} is disjoint from W:=\bigcup_{i=1}^n W_{q_i} (verify this through the algebra of sets), and since K\subset W we have V\subseteq K^c a neighborhood of p. So p is an interior point, and since p\in K^c is arbitrary, all points in K^c are interior---which is equivalent to saying that K^c is open.

    EDIT: You have several typos in your OP. For example the theorem should read "compact subsets of metric spaces are closed." And your LaTeX code is off sometimes. Please note that not everyone can see through to what you mean.
    Last edited by hatsoff; July 18th 2011 at 11:56 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,798
    Thanks
    1690
    Awards
    1

    Re: Understanding Topology Proof

    Quote Originally Posted by I-Think View Post
    Studying on my own currently and having trouble understanding a topology proof.

    Theorem: Compact subsets of metric spaces are compact
    There is nothing there to prove. Compact subsets of any space are compact.

    There is a theorem that says: Closed subsets of a compact space are compact.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,989
    Thanks
    1652

    Re: Understanding Topology Proof

    There are also "compact subsets of a metric space are bounded" and "compact subsets of a metric space (actually any topological space) are closed." Please go back and reread the problem.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7

    Re: Understanding Topology Proof

    The proof obviously shows that compact subsets of metric spaces are closed. So presumably that is what the theorem should be stating.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member I-Think's Avatar
    Joined
    Apr 2009
    Posts
    288

    Re: Understanding Topology Proof

    I apologize for typos present. I hope it's more readable now
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: Understanding Topology Proof

    Quote Originally Posted by HallsofIvy View Post
    "compact subsets of a metric space (actually any topological space) are closed."
    Well........
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Senior Member Tinyboss's Avatar
    Joined
    Jul 2008
    Posts
    433

    Re: Understanding Topology Proof

    I'd approach this one by contraposition: prove that if a subset S of a metric space is not closed, then S is not compact, by exhibiting a cover with no finite subcover.

    Since S isn't closed, there is a point x which is a limit point of S, but not contained in S. So let your open cover be complements of closed 1/n balls around x.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Not Understanding Proof for Derivative.
    Posted in the Calculus Forum
    Replies: 8
    Last Post: January 14th 2011, 04:07 AM
  2. Help understanding proof
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 9th 2010, 10:58 AM
  3. Help Understanding a Proof
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 23rd 2009, 08:37 PM
  4. Help understanding this theorem/proof
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: December 8th 2009, 03:39 PM
  5. trouble understanding the relative topology and subspaces
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: June 5th 2008, 09:38 PM

Search Tags


/mathhelpforum @mathhelpforum