Let us define the boxing dimension $\text{bim}(S)$ of a set $S \subseteq \mathbb Z$ as the smallest integer $k \ge 0$ for which there is a family of $k$ discrete intervals (that is, intervals of the poset of integers with their usual ordering) that cover $S$ (in the sense that $S$ is contained in their union), with the understanding that if no such integer $k$ exists then $\text{bim}(S) := \infty$.

The boxing dimension of the set $S$ is zero if and only if $S$ is empty, and it is one if and only if $S$ is a discrete interval. More generally, the boxing dimension of $S$ is equal to a certain integer $k \ge 0$ if and only if there is a unique way to decompose $S$ as a union of $k$ well-separated non-empty discrete intervals, where we say that two sets $X, Y \subseteq \mathbb Z$ are well separated if $|x-y| \ge 2$ for all $x \in X$ and $y \in Y$ (so, for instance, $\{0, 5\}$ and $\{-2, 2\}$ are well-separated, but $\{0, 5\}$ and $\{-1, 2\}$ are not).

MY QUESTIONS. (1) Does (what I'm calling) the boxing dimension have a more standard name? Or is it a special case of a more general standard notion? (2) Likewise, do well-separated subsets of $\mathbb Z$ have a more standard name? Or are they a special case of a more general standard notion?

For what it's worth, my interest in this notion comes from a joint project with an undergraduate student where the classification of certain automorphism groups boils down to a (somewhat unusual) induction on the boxing dimension of certain subsets of $\mathbb Z$.