I have a question
Let $d$ be a metric on $X$, and define the set to set distance
$$\operatorname{dist}(A,B) = \inf\{d(x,y): x\in A, y \in B\}$$
where $A,B \subseteq X$ are nonempty sets
Show that $A\cap B \neq \varnothing \Rightarrow \operatorname{dist}(A,B) = 0$, and $\operatorname{dist}(A, B) = 0 \not\Rightarrow A\cap B \neq \varnothing$
First: ($A\cap B \neq \varnothing \Rightarrow \operatorname{dist}(A,B) = 0$)
Trivial, since $A \cap B \neq \varnothing \implies \exists z \in A$ and $B$, so $\operatorname{dist}(A,B) = \inf\{d(z,z)\} = 0$
Second: ($\operatorname{dist}(A, B) = 0 \not\Rightarrow A\cap B \neq \varnothing$)
We want to produce $A \cap B = \varnothing$ such that $\operatorname{dist}(A,B) = 0$. Is there a metric space where this can happen?
I've checked the discrete metric, all the $\ell_p$ metrics. I don't think you can have disjoint sets with their distance zero.