MHB Problem of the Week #120 - September 15th, 2014

[Chris L T521]

Here's this week's problem!

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Problem: Let $A$ and $B$ be subsets of a metric space $(X,\rho)$. Define
\[\mathrm{dist}(A,B) = \inf\{\rho(u,v)\mid u\in A,\,v\in B\}.\]
If $A$ is compact and $B$ is closed, show that $A\cap B=\emptyset$ if and only if $\mathrm{dist}(A,B)>0$.

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[Chris L T521]

This week's problem was correctly answered by Ackbach and Euge. You can find Euge's solution below.

[sp]Suppose $A\cap B \neq \emptyset$. Then there is an $x \in X$ such that $x\in A$ and $x\in B$; hence, $0 = \rho(x,x) \le \text{dist}(A,B)$. So, $\text{dist}(A,B) = 0$. Conversely, suppose $A\cap B = \emptyset$. The mapping from $A$ to $\Bbb R$ given by $a \mapsto \rho(a,B)$ is continuous on the compact set $A$, so it has a minimum value at some $a_0 \in A$. If $\rho(a_0, B) = 0$, then since $B$ is closed, $a_0 \in B$, contradicting the assumption $A\cap B = \emptyset$. Hence, $\rho(a_0, B) > 0$. For all $a\in A$ and $b\in B$, $\rho(a,b) \ge \rho(a, B) \ge \rho(a_0, B)$. Therefore, $\text{dist}(A,B) \ge \rho(a_0, B) > 0$.[/sp]