Results 1 to 4 of 4

Math Help - Metric Spaces ...... help!

  1. #1
    Junior Member
    Joined
    Dec 2009
    Posts
    30

    Wink Metric Spaces ...... help!

    Please help me in finding a possible solution for this problem that doesn't involve the whole space or the empty set .... b/c my professor made that requirement.

    Problem.
    Give an example of a non-empty space X with metric d, in which every closed set is the intersection of a non-empty family of closed balls.

    Please Help! Show all work
    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 derek walcott View Post
    Problem.
    Give an example of a non-empty space X with metric d, in which every closed set is the intersection of a non-empty family of closed balls.
    You could take X to be a space consisting of just two points, say p and q, with d(p,q) = 1 (and d(p,p) = d(q,q) = 0, of course).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Dec 2009
    Posts
    30

    Cool

    okay. i did consider the simple space with only two elements but i am puzzled on the closed balls.

    i know that a closed ball is C(a) ={x in X: d(a,x) <=r}
    but i don't have any representation of what it actually looks like? can it leave the set? or am i missing the obvious and is the radius = to every point in between and including 1?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member qspeechc's Avatar
    Joined
    Oct 2007
    Posts
    73
    The entire space is the two points p and q, so your question of whether it leaves the set makes no sense.
    The closed balls of radius less than one consists of one point, and the closed balls or radius at least one consists of both points. This is all the closed balls.
    The closed sets are: the empty set, {p}, {q} and {p,q}=X.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Any metric spaces can be viewed as a subset of normed spaces
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: December 15th 2011, 03:00 PM
  2. [SOLVED] metric spaces
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: October 30th 2010, 10:55 AM
  3. metric spaces
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 20th 2010, 11:11 AM
  4. Metric spaces
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: September 3rd 2010, 08:26 AM
  5. metric spaces
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 3rd 2009, 12:33 AM

Search Tags


/mathhelpforum @mathhelpforum