is\ bounded\ on\ } T \}.\end{aligned}$$ Other metrics on $X$ can define the same coarse structure. In coarse geometry, we focus on features which only depends on the coarse structure $\mathcal{C}$.

A coarse space $(X, \mathcal{C})$ is said to be uniformly locally finite if every controlled set $T \in \mathcal{C}$ satisfies $\sup_{x \in X} \sharp(T[x]) < \infty$.

Let us observe what uniform local finiteness means in the case of metric spaces. For a metric space $(X, d)$, we use the following notations: $$\begin{aligned}
N_R(Y) &=& \{x \in X \ |\ d(x, Y) \le R \}, \\
\partial_R (Y) &=& N_R(Y) \setminus Y.\end{aligned}$$ A metric space $(X, d)$ is uniformly locally finite if and only if for every distance $R > 0$, all the balls $N_R(x)$ of radius $R$ are finite and the numbers of their elements are uniformly bounded, namely, $\mathrm{sup}_{x \in X} \sharp(N_R(x)) < \infty$. In many references, metric spaces with this property are said to have bounded geometry.

The other typical example of a coarse space arises from groups and group actions. Let $G$ be a discrete group. A set $X$ equipped with a $G$-action naturally has a coarse structure. For a finite subset $K \subseteq G$, define 