1. Not finding help here? Sign up for a free 30min 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!

Distance between Sets and their Closures

  1. Oct 11, 2011 #1
    1. The problem statement, all variables and given/known data

    Suppose [itex](X,d)[/itex] is a metric space, and suppose that [itex]A,B\subseteq X[/itex]. Show that [itex]dist(A,B)=dist(cl(A),cl(B))[/itex].

    2. Relevant equations
    [itex]cl(A)=\partial A\cup A[/itex].
    [itex]dist(A,B)=\inf \{d(a,b):a\in A,b\in B\}[/itex]


    3. The attempt at a solution
    Its clear that [itex]dist(cl(A),cl(B))\leq \min\{dist(A,B),dist(A,\partial B),dist(\partial A,B),dist(\partial A,\partial B)\}\leq dist(A,B)[/itex]. I just can't exactly find a way to go the other direction of the inequality. I'm not looking for a direct solution, just some intuition.
     
  2. jcsd
  3. Oct 11, 2011 #2
    Here's my attempt at a solution:

    [itex]dist(A,B)\leq dist(A,x)+dist(x,B)\implies \inf _{x\in cl(A)}\{dist(A,B)\}\leq \inf _{x\in cl(A)}\{dist(A,x)+dist(x,B)\}=\inf _{x\in cl(A)}\{dist(x,B)\}[/itex]. And, [itex]dist(cl(A),B)\leq dist(cl(A),y)+dist(y,B)\implies \inf _{y\in cl(B)}\{dist(cl(A),B)\}\leq\inf _{y\in cl(B)}\{dist(cl(A),y)+dist(y,B)\}=\inf _{y\in cl(B)}\{dist(cl(A),y)\}=dist(cl(A),cl(B))[/itex]. Hence, [tex]dist(A,B)=\inf _{x\in cl(A)}\{\inf _{y\in cl(B)}\{dist(A,B)\}\}\leq dist(cl(A),cl(B)).[/tex]

    Does this work?
     
  4. Oct 11, 2011 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    dist(cl(A),cl(B)) is clearly less than dist(A,B) just because A is contained in cl(A) and B is contained in cl(B). Do you agree with that? If so why? To go farther you need to deal with the definition of what cl(A) or boundary(A) in a metric space is. What is it?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Distance between Sets and their Closures
  1. Closure of set (Replies: 19)

Loading...