For a one-point set $\{x\}$, we simply write $T[x] = T[\{ x \}]$. A subset $F \subseteq X$ is called a $T$-bounded set if there exists $x \in X$ such that $F \subseteq T[x]$.

\[Definition 2.3 in [@RoeLectureNote]\] Let $X$ be a set and let $\mathcal{C}$ be a family of subsets of $X^2$. The pair $(X, \mathcal{C})$ is said to be a coarse space if it satisfies the following:

-   The diagonal subset $\Delta_X \subseteq X^2$ is an element of $\mathcal{C}$.

-   If $T_1 \subseteq T_2$ and $T_2 \in \mathcal{C}$, then $T_1 \in \mathcal{C}$.

-   If $T \in \mathcal{C}$, then $T^{-1} \in \mathcal{C}$.

-   If $T_1, T_2 \in \mathcal{C}$, then $T_1 \circ T_2 \in \mathcal{C}$.

-   If $T_1, T_2 \in \mathcal{C}$, then $T_1 \cup T_2 \in \mathcal{C}$.

Two elements $x, y \in X$ are said to be connected if $\{(x, y)\} \in \mathcal{C}$. Sometimes, a coarse space is assumed to be connected. In this paper, connectivity is not required. For the coarse space $(X, \mathcal{C})$, elements of $\mathcal{C}$ are called controlled sets (or entourages).

A metric space $(X, d)$ is naturally equipped with a coarse structure $\mathcal{C}$ defined by $$\begin{aligned}
\mathcal{C} = \{ T \subseteq X^2 \ |\ d \mathrm{\ 