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Let $A$ and $B$ be two sets of real numbers. Define the distance from $A$ to $B$ by $$\rho (A,B) = \inf \{ |a-b| : a \in A, b \in B\} \;.$$ Give an example to show that the distance between two closed sets can be $0$ even if the two sets are disjoint.

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Exercise for you: show that if this is true, $A$ or $B$ must be non-compact. –  Neal May 16 '12 at 16:42

3 Answers 3

up vote 10 down vote accepted

Let $A = \mathbb N$ and let $B = \left\{n+\frac{1}{2n} :n\in \mathbb N\right\}$. Then A and B are closed and disjoint, but $$\inf \{|a−b|:a \in A,b \in B\} = \inf \left | \frac{1}{2n}\right| = 0$$

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Hi. You may want to see some of the TeX edits I have made by clicking on the time stamp above my name. –  user21436 Mar 29 '12 at 19:16
    
And you prove that it is closed because it's the complement of an infinite amount of open intervals? –  Thomas Ahle Aug 13 at 14:46

Consider the sets $\mathbb N$ and $\mathbb N\pi = \{n\pi : n\in\mathbb N\}$. Then $\mathbb N\cap \mathbb N\pi=\emptyset$ as $\pi$ is irrational, but we have points in $\mathbb N\pi$ which lie arbitrarily close to the integers.

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This is a nice example, but showing that $\mathbb{N}\pi$ is arbitrarily close to $\mathbb{N}$ is probably harder than the simplest example to original question :). –  Theo May 16 '12 at 16:06

Let $A$ be the set of positive integers, and let $B$ be the set of all numbers of the form $n+1/n$, where $n$ ranges over the integers $\gt 1$.

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For $n=1$ you get $n+1/n=2$ - that's why in another answer $n+\frac1{2n}$ was taken. (Or you cut simply add the condition $n\ge 2$.) –  Martin Sleziak May 16 '12 at 14:24
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@MartinSleziak: Thanks for spotting this. I had forgotten that $1+1=2$. –  André Nicolas May 16 '12 at 14:28

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