Results 1 to 4 of 4

Math Help - Topology and metrics

  1. #1
    Newbie
    Joined
    Feb 2011
    Posts
    22

    Topology and metrics

    find two metric functions (distance) d1 , d2 on the space V=(0,1) (the interval 0,1).
    d1 , d2 must support:
    a. V is complete with the metric d1 and incomplete with d2.
    b. d1 , d2 induce the same topology on V (same topological space).

    I apologize for any spelling mistakes and appreciate your help
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by aharonidan View Post
    find two metric functions (distance) d1 , d2 on the space V=(0,1) (the interval 0,1).
    d1 , d2 must support:
    a. V is complete with the metric d1 and incomplete with d2.
    b. d1 , d2 induce the same topology on V (same topological space).
    With the usual metric d_1, where d_1(x,y) = |x-y|, V is incomplete. So the problem is to find another metric d_2, under which V is complete. Use the fact that that the interval (0,1) is homeomorphic to the whole real line. The homeomorphism can be used to transport the usual metric on \mathbb{R} across to the unit interval, thereby giving it a complete metric.

    Specifically, if 0,1)\to \mathbb{R}" alt="f0,1)\to \mathbb{R}" /> is a homeomorphism, then you can take d_2(x,y) = |f(x)-f(y)|.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Feb 2011
    Posts
    22
    thanks for the help.
    I wasn't clear enough or fully understood your soulution.
    the functions (d1 , d2) must implement both a and b.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by aharonidan View Post
    thanks for the help.
    I wasn't clear enough or fully understood your solution.
    the functions (d1 , d2) must implement both a and b.
    The functions d_1 and d_2 that I suggested do implement both a and b, apart from the fact that I had them the wrong way round. My d_1 makes the space incomplete and d_2 makes it complete. You wanted it the other way round, but I'm sure you can cope with that.

    Take my d_1 first. It is the usual metric on (0,1), and it induces the usual topology. The space is not complete with this metric because for example the sequence (1/n) is Cauchy but does not converge to a point in the space.

    The other metric is a bit more complicated. I suggested choosing a function f from (0,1) to \mathbb{R} that is invertible and such that f and its inverse are both continuous. There are many such functions, for example f(t) = \tan\bigl((t-\frac12)\pi\bigr). Now define d_2(s,t) = |f(s)-f(t)|. Check first that this is a metric on (0,1). Then notice that it induces the usual topology on the unit interval – that follows from the fact that f and its inverse are continuous.

    Finally, the interval (0,1) is complete for the metric d_2. The reason for that is that if (t_n) is Cauchy for the d_2 metric then (f(t_n)) will be Cauchy for the usual metric on \mathbb{R}. But \mathbb{R} with its usual metric is complete and therefore the sequence (f(t_n)) converges to some point x. Thus d_2\bigl(t_n,f^{-1}(x)\bigr) = |f(t_n) - x|\to0, in other words (t_n) converges in the d_2 metric.

    I hope that makes things clearer.
    Last edited by Opalg; March 2nd 2011 at 12:45 PM. Reason: corrected error
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Fun with metrics
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: September 24th 2011, 07:05 AM
  2. Equivalent Metrics
    Posted in the Differential Geometry Forum
    Replies: 10
    Last Post: February 16th 2010, 03:24 AM
  3. Discrete metrics (basic topology)
    Posted in the Differential Geometry Forum
    Replies: 10
    Last Post: January 7th 2010, 07:40 AM
  4. metrics
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: December 10th 2009, 09:29 AM
  5. Metrics
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 29th 2008, 11:38 AM

Search Tags


/mathhelpforum @mathhelpforum