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Proof or give counterexample:

Let $(M,d)$ be a metric space, $A$ a closed and $B$ a compact subset with $A\cap B\neq\emptyset$. Then $$\inf \{d(a,b): a\in A\setminus B\land\ b\in B\setminus A \}>0.$$

For disjoint sets the proof is easy, but in this case I don't even know if it is true.

mag
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1 Answers1

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This is not true take $A=[0,1], B=[1,2]$, $A\cap B=\{1\}$ $d(1-{1\over n},1+{1\over n})={2\over n}$, $n>1$ is an integer.

Tsemo Aristide
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