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Closed set, compact set, and a definition of distance between sets

  1. Oct 13, 2009 #1

    txy

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    1. The problem statement, all variables and given/known data

    Let E and F be 2 non-empty subsets of [tex]R^{n}[/tex]. Define the distance between E and F as follows:

    d(E,F) = [tex]inf_{x\in E , y\in F} | x - y |[/tex]

    (a). Give an example of 2 closed sets E and F (which are non-empty subsets of R^n) that satisfy d(E,F) = 0 but the intersection of E and F is a null set.

    (b). If E and F (non-empty subsets of R^n) are compact sets and d(E,F) = 0, prove that the intersection of E and F cannot be the null set.


    2. Relevant equations

    3. The attempt at a solution

    Part (b) of the question suggests that the closed sets in part (a) are not bounded. But I still can't find these 2 sets with d(E,F) = 0 and yet they don't intersect.

    For part (b), maybe I can find a sequence of points [tex]x_{k}[/tex] in E and another sequence [tex]y_{k}[/tex] in F such that as k increases, the distance between [tex]x_{k}[/tex] and [tex]y_{k}[/tex] decreases. Then since E and F are compact, therefore these 2 sequences must converge. And since d(E,F) = 0, hence these 2 sequences must converge to the same point. And then since E and F are compact, this limit point must lie in both E and F, thus E and F share at least one common point.
     
  2. jcsd
  3. Oct 13, 2009 #2

    Office_Shredder

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    For part (b), think about the graphs of 1/x2 and -1/x2 as an example.

    That looks like a pretty good sketch argument. Only thing is that only a subsequence of each of those must converge, but it amounts to the same result here
     
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