Results 1 to 4 of 4
Like Tree3Thanks
  • 2 Post By Plato
  • 1 Post By Plato

Math Help - Equality of two balls in a metric space

  1. #1
    Member
    Joined
    Sep 2012
    From
    india
    Posts
    78

    Equality of two balls in a metric space

    Is it posssible for b[x:r) and b[y;s) to be equal with x not equal to y and r not equal to s ?
    I know it is possible,for instance if we consider a non empty set X with the discrete metric, then for each x in X the balls b[x;r) for r in (0,1] are equal to the singleton set {x}. Also the balls b[x;r) for r in (1,infinity) are equal to X for all x in X.
    What is the idea behind two balls with different radii and centre's being equal ?
    What i don't understand is, that even in the above example, in what sense are the two balls equal ?
    What is the meaning of equality of two balls in a metric space ?
    In this example one ball has only singleton element {x} and the other one is the whole metric space X then how are they equal ?

    I am a little confused !
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1

    Re: Equality of two balls in a metric space

    Quote Originally Posted by mrmaaza123 View Post
    Is it posssible for b[x:r) and b[y;s) to be equal with x not equal to y and r not equal to s ? I know it is possible,for instance if we consider a non empty set X with the discrete metric, then for each x in X the balls b[x;r) for r in (0,1] are equal to the singleton set {x}. Also the balls b[x;r) for r in (1,infinity) are equal to X for all x in X.
    What is the idea behind two balls with different radii and centre's being equal ?
    Those of trained in the tradition of R L Moore are distrustful of empty point sets.
    One of the most basic properties of metric is: if x\ne y then d(x,y)>0.
    Now if x\ne y then let r=\frac{d(x,y)}{2}>0.

    Then it should be very clear that \mathfrak{B}\left( {x;r} \right) \cap \mathfrak{B}\left( {y;r} \right) = \emptyset .

    How could they be equal ?
    Thanks from topsquark and mrmaaza123
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Sep 2012
    From
    india
    Posts
    78

    Re: Equality of two balls in a metric space

    The interval in which r lies is changing so isn't "r" changing ? How can we take it to be the same for both the cases ?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1

    Re: Equality of two balls in a metric space

    Quote Originally Posted by mrmaaza123 View Post
    The interval in which r lies is changing so isn't "r" changing ? How can we take it to be the same for both the cases ?
    Please read this page on Hausdorff spaces.
    Every metric space is a Hausdorff space. So points are separated.
    Balls are determined by first a point and a positive real number.
    This if balls are equal the the centers are the same.
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: July 8th 2011, 02:16 PM
  2. Limit of function from one metric space to another metric space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 17th 2010, 02:04 PM
  3. a basket contains 5 red balls, 3 blue balls, 1 green balls
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: May 28th 2010, 02:39 AM
  4. Sets > Metric Space > Euclidean Space
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 25th 2010, 10:17 PM
  5. Open balls in metric spaces
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: February 12th 2009, 01:23 PM

Search Tags


/mathhelpforum @mathhelpforum