1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Finite subset of a metric space

  1. Feb 24, 2008 #1
    1. The problem statement, all variables and given/known data

    Let X be an infinite set. For p,q [tex]\in[/tex] X define:
    d(p,q) = {1 if p [tex]\neq[/tex] q; 0 if p = q
    Suppose E is a finite subset of X, find all limit points of E.

    2. Relevant equations

    definition: a point p is a limit point of E if every neighborhood of p contains a point q [tex]\neq[/tex] p s.t. q[tex]\in[/tex]E

    3. The attempt at a solution
    My thoughts were that there are no limit points for any subset of X because for any point p, B(p, 1/n) is empty, [tex]\forall[/tex]n[tex]\in[/tex]N. Therefore, for any point p, there exists at least one neighborhood that contains no points q [tex]\in[/tex] E.

    I feel that this is true, but have I made any incorrect assumptions??
    Thanks a lot for any feedback.
     
  2. jcsd
  3. Feb 24, 2008 #2

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    B(p,1/n) isn't empty for all n. For instance p is always in B(p,1/n), and if n=1, then B(p,1/n) is in fact all of X.

    You definitely have the right idea, but you have to be a bit more careful.
     
  4. Feb 24, 2008 #3

    HallsofIvy

    User Avatar
    Science Advisor

    remember that you have the additional rquirement that q[itex]\ne[/itex] p. I think that is what you meant to say. For all n, B(p, 1/n) contains p, but for some n, B(p, 1/n) does not include any other point of the set.
     
  5. Feb 25, 2008 #4
    Right, ok. I can fix that detail. Thanks a lot guys.
     
  6. Sep 1, 2011 #5
    i need the proof for every finite set is a closed set in a metric space ..
    and finite set in a metric space in compact as soon as possible../.
     
  7. Sep 1, 2011 #6
    You will need to make a new topic for this. And it would also be nice to see your attempt...
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...