Distance between two disjoint closed subsets of the real numbers

Question

Let $A$ and $B$ be disjoint closed subsets of $\mathbb R^n$. Define $d(A,B)=\inf\{\left\|a-b\right\|\colon a \in A \text{ and } b \in B\}$.

I have to show that if $A=\{a\}$ is a singleton, then $d(A,B) > 0$.

Also I am not sure how to show that if $A$ is compact, $d(A,B) > 0$.

Also I don't know how to find an example of two disjoint closed sets in $\mathbb R^2$ with $d(A,B) =0$.

Hint: suppose $A={a}$ is a singleton, and d(A,B)=0. Then we can pick a sequence $b_n \in B$ such that $d(a,b_n)$ tends to 0 as n tends to infinity. Conclude from this that $a \in B$. That's a contradiction (why?). So we must have d(A,B) > 0. — jflipp, Oct 30 '14 at 08:34

score 0 · Answer 1 · edited Apr 13 '17 at 12:19

0

For your first question, please see jfilpp's comment.

For your second question, i.e. $A$ is compact, see here.

Here is an example for your last question: $A=\mathbb{R}\times\{0\},B=\{(x,y)|xy=1\}$. These are closed disjoint sets, but $dist(A,B)=0$ (consider $(n,0)\in A, (n,\frac{1}{n})\in B$ ).

edited Apr 13 '17 at 12:19

Community

1

answered Oct 30 '14 at 10:10

John

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Distance between two disjoint closed subsets of the real numbers

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