theorem, this groupoid is too big. We need separability of translation C$^*$-algebra and the groupoid must be second countable. In this section, we first construct a C$^*$-algebra, which is associated to a finite collection of partial bijections. We describe the algebra as a representation of a groupoid, whose topology is generated by countably many closed and open subsets.

Let $X$ be a uniformly locally finite coarse space. For a controlled set $T$, there exists a label $L$ in the following sense.

Let $T$ be a controlled set containing the diagonal subset $\Delta_X \subseteq T \subseteq X^2$. Label on $T$ is a family $L = \{\Delta_X = \phi(0), \phi(1), \phi(2), \cdots, \phi(k) \}$ of subsets of $T$ satisfying the following conditions:

-   $T = \Delta_X \bigcup \left(\bigcup_{i = 1}^k \phi(i) \right)$,

-   Each $\phi(i)$ gives a bijection from a subset of $X$ to a subset of $X$. Namely, if $(z, x), (z, y) \in \phi(i)$, then $x = y$, if $(x, z), (y, z) \in \phi(i)$, then $x = y$.

When $(x, y) \in X^2$ is an element of $\phi(i)$, we regard $x$ as the image of $y$ with respect to the map $\phi(i)$. Then we write $x = [\phi(i)](y)$. Let 