Results 1 to 4 of 4

Math Help - Compactness over discrete metric space

  1. #1
    Newbie
    Joined
    Mar 2011
    Posts
    9

    Compactness over discrete metric space

    Any idea that help in solving this problem is highly appreciated
    Attached Thumbnails Attached Thumbnails Compactness over discrete metric space-untitled.jpg  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,708
    Thanks
    1638
    Awards
    1

    Re: Compactness over discrete metric space

    Question:
    Is \left\{ {\mathfrak{B}\left( {x;0.5} \right):x \in [0,1]} \right\} an open covering of [0,1]~?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2011
    Posts
    9

    Re: Compactness over discrete metric space

    Thanks for your reply

    So you are saying that all the points are isolated point so we don't have a finite open cover of the set ?
    Also, there is a theorem that states that every bounded and closed set in R^k is compact. so is the set of isolated point closed
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,708
    Thanks
    1638
    Awards
    1

    Re: Compactness over discrete metric space

    Quote Originally Posted by mia25 View Post
    So you are saying that all the points are isolated point so we don't have a finite open cover of the set ?
    Correct.

    Quote Originally Posted by mia25 View Post
    Also, there is a theorem that states that every bounded and closed set in R^k is compact.
    But that theorem is not for R^k with a discrete metric.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. The discrete Metric- Completeness and Compactness
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: October 10th 2010, 01:50 AM
  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. discrete metric space n singleton sets
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 21st 2010, 05:59 PM
  4. Compactness and metric space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: May 5th 2009, 12:28 PM
  5. Compactness in a metric space
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 17th 2007, 12:14 AM

Search Tags


/mathhelpforum @mathhelpforum