Example of two closed disjoint set $X, Y$ so that $d(X, Y) = 0$

Question

I am looking for an example of a metric space $M$ and non empty disjoint closed subsets $X$ and $Y$ such I that $d(X,Y)=0$, where $$d(X,Y)=\inf_{x\in X, y\in Y} d(x, y).$$

I’m thinking it might have to do with the discrete metric, but I cannot wrap my head around this since they are disjoint.

Answer 1

In $M = \mathbb R^2$, as suggested in the comment, one can take $X = \{xy = 0\}$ and $Y = \{xy = 1\}$. They are infinitely closed to each other when $(x, y) \to \infty$.

One can also take an example in $\mathbb R$, letting $X = \mathbb N$ and $Y = \{ n + \frac 1{2n} : n\in \mathbb N\}$.

On the other hand, one cannot find such an example in a discrete metric space, as $d(X, Y) = 1$ whenever $X, Y$ are disjoint.