Page 2 of 2 FirstFirst 12
Results 16 to 20 of 20

Math Help - COMPLETE metric space

  1. #16
    Senior Member
    Joined
    Jan 2009
    Posts
    404
    Quote Originally Posted by Drexel28 View Post
    If the radii of the sets does not approach zero then it must contain more elements than a sequence of sets whose radius approaches zero...so??
    Let
    d(m,n)= 1/2 + ∑1/(2^k) where the sum is from k=m to k=n-1, if m<n
    d(m,n)=0, if m=n
    d(m,n)=d(n,m), if m>n

    d(1,2)=1, d(2,3)=0.75, d(3,4)=0.625
    Define
    B1=Closed ball of radius 1 about 2={1,2,3,4,...}
    B2=Closed ball of radius 0.75 about 3={2,3,4,...}
    B3=Closed ball of radius 0.625 about 4={3,4,5,...} <---Is this correct? Could someone confirm this?
    ...

    If so, then it's a decreasing sequence of closed balls with empty intersection.

    Now the problem is whether this metric space (N,d) is complete or not. (complete means every Cauchy sequence in N converges (in N))
    For any n E N, the open ball of radius 1/2 about n = B(1/2,n) = {n}. But does this show that every Cauchy sequence in N converges (in N)) ??? I'm puzzled about this part, and I would appreciate if someone can help me with this part.

    Thanks for any help!
    Last edited by kingwinner; February 8th 2010 at 08:16 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #17
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by kingwinner View Post
    Let
    d(m,n)= 1/2 + ∑1/(2^k) where the sum is from k=m to k=n-1, if m<n
    d(m,n)=0, if m=n
    d(m,n)=d(n,m), if m>n

    d(1,2)=1, d(2,3)=0.75, d(3,4)=0.625
    Define
    B1=Closed ball of radius 0.1 about 2={1,2,3,4,...}
    B2=Closed ball of radius 0.75 about 3={2,3,4,...}
    B3=Closed ball of radius 0.625 about 3={3,4,5,...} <---Is this correct? Could someone confirm this?
    ...

    If so, then it's a decreasing sequence of closed balls with empty intersection.

    Now the problem is whether this metric space (N,d) is complete or not. (complete means every Cauchy sequence in N converges (in N))
    For any n E N, the open ball of radius 1/2 about n = B(1/2,n) = {n}. But does this show that every Cauchy sequence in N converges (in N)) ??? I'm puzzled about this part, and I would appreciate if someone can help me with this part.

    Thanks for any help!
    I'm telling you. In a complete metric space a decreasing sequence of sets has non-empty intersection. If it's diameter does not approach zero then it must contain MORE elements then if it's diameter approaches zero. The question is incorrect as far as I can see.
    Follow Math Help Forum on Facebook and Google+

  3. #18
    Senior Member
    Joined
    Jan 2009
    Posts
    404
    The following satisfies every single requirement in the original question, so it is a valid counterexample.

    Define the metric
    d(m,n)= 1/2 + ∑1/(2^k) where the sum is from k=m to k=n-1, if m<n
    d(m,n)=0, if m=n
    d(m,n)=d(n,m), if m>n

    d(1,2)=1, d(2,3)=0.75, d(3,4)=0.625
    Define
    B1=Closed ball of radius 1 about 2={1,2,3,4,...}
    B2=Closed ball of radius 0.75 about 3={2,3,4,...}
    B3=Closed ball of radius 0.625 about 4={3,4,5,...}
    ...etc

    This is a decreasing sequence of closed balls with empty intersection.

    Claim: the metric space (N,d) is complete. (complete means every Cauchy sequence in N converges (in N))
    Proof:
    For any n E N, the open ball of radius 1/2 about n = B(1/2,n) = {n}.
    Let {a_n} be any Cauchy sequence in N, then there exists K s.t. d(a_n,a_m)<1/4 for all n,m>K.
    => for all n,m>K, a_n=a_m
    => every Cauchy sequence in N is eventually constant, and hence converges. Thus (N,d) is complete.

    I think every step is properly justified, so this is indeed a counterexample. Please feel free to point out any mistakes if there is one.
    Last edited by kingwinner; February 9th 2010 at 09:09 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #19
    Junior Member
    Joined
    Feb 2010
    From
    Lisbon
    Posts
    51
    I think your example is indeed correct. Sorry for misleading you with my other reply. I said "in a compact space (in this case, the first ball), any intersection of non-empty nested closed sets is non-empty.". The statement is true, but... a closed ball need not be compact xD Sorry for that And good job ^^
    Follow Math Help Forum on Facebook and Google+

  5. #20
    Senior Member
    Joined
    Jan 2009
    Posts
    404
    Quote Originally Posted by Nyrox View Post
    I think your example is indeed correct. Sorry for misleading you with my other reply. I said "in a compact space (in this case, the first ball), any intersection of non-empty nested closed sets is non-empty.". The statement is true, but... a closed ball need not be compact xD Sorry for that And good job ^^
    Thanks for confirming!

    There is a theorem that says: a metric space is complete iff every decreasing sequence of closed ball with radii going to zero has nonempty intersection.

    And as Drexel28 pointed out, if it's diameter does not approach zero then it must contain MORE elements then if it's diameter approaches zero. This makes perfect sense to me as well.

    But I indeed have constructed a concrete counterexample.

    What explains this inconsistency?
    Follow Math Help Forum on Facebook and Google+

Page 2 of 2 FirstFirst 12

Similar Math Help Forum Discussions

  1. Surjectivity of an Isometry given the metric space is complete.
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: March 6th 2012, 10:23 PM
  2. A complete Metric Space
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: April 7th 2011, 12:43 PM
  3. Complete metric space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 12th 2011, 08:46 PM
  4. Complete metric space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: May 3rd 2010, 04:23 PM
  5. Complete metric space
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 16th 2009, 11:26 PM

Search Tags


/mathhelpforum @mathhelpforum