I am trying to solve the following problems.
Find a metric space $X$ and two disjoint closed subsets $E,F$ such that $$ d(E,F) = \inf \{d(x,y) \mid x \in E, \; y \in F\} $$ is equal to $0$.
I first tried the discrete metric on some infinite set $X$. This doesn't work, because if $E$ and $F$ are disjoint, then $d(x,y) = 1$ for any $(x,y) \in E \times F$. I think using a "discrete" set is intended because such a set has no limit points and is therefore closed. If it's a subset of $\mathbb{R}^k$, then it would have to be unbounded. Otherwise, it would be compact and this statement would be false. Unbounded intervals in $\mathbb{R}^1$ has the form $[a, \infty)$ and $(-\infty, a]$. There doesn't seem to be a way to pick those so that they are disjoint.
Consider the metric space of rational numbers, $A = \mathbb Q \cap (-\infty, \pi], B = \mathbb Q \cap [\pi,\infty)$. These are closed because the complements are open rational intervals. But since the rationals are dense in the reals, both of these sets get arbitrarily close to $\pi$, thus to each other.