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

1. Oct 13, 2009

### txy

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

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

d(E,F) = $$inf_{x\in E , y\in F} | x - y |$$

(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 $$x_{k}$$ in E and another sequence $$y_{k}$$ in F such that as k increases, the distance between $$x_{k}$$ and $$y_{k}$$ 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. Oct 13, 2009

### Office_Shredder

Staff Emeritus
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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