I am reading a paper about property (T) for groupoids: Topological property (T) for groupoids. In section 4.4 they discuss the HLS groupoids which I describe define here.

Let $\Gamma$ be a discrete group and $(N_k)_{k \in \mathbb N}$ a decreasing sequence of normal subgroups of finite index. Let $\Gamma_k := \Gamma/N_k$ and denote $q_k: \Gamma \rightarrow \Gamma_k$ the quotient homomorphism. If we let $\overline{\mathbb N}$ to be the one-point compactification of $\mathbb N$ we define $G$ to be the quotient of $\overline{\mathbb N} \times \Gamma$ by the relation $$ (k,t) \sim (l,u) \, \text{ if } \, k = l \, \text{ and } \, q_k(t) = q_k(u), $$ Then $G$ is a group bundle over $\overline{\mathbb N}$, meaning it is a topological groupoid (étale and even ample) for the obvious structure maps. We further assume $G$ is Hausdorff. For every $t \in \Gamma$, define the set $$ S_t := \{[k,q_n(t)]: n \in \mathbb N \} $$ In Lemma 4.10, they claim that the assignment $t \mapsto \chi_{S_t}$ extends to an injective $*$-homomorphism $\iota: \mathbb C\Gamma \rightarrow C_c(G)$.

I don't think this map is well defined since $\chi_{S_t}$ is not necessarily continuous on $G$. Am I seeing something wrong? Or is this a typo and they meant $\overline{\mathbb N}$ instead of $\mathbb N$? They consistently use $\mathbb N$ which makes me believe it is hard to be a typo.

Remark: Their main use of this Lemma is to prove Proposition 4.11 which relates the Property (T) of $G$ with that of $\Gamma$. Somewhere in the proof they use the fact that the $S_t$'s are compact (I think they need the set $K$ defined in that proposition to be compact), but this only only true if we consider $\overline{\mathbb N}$ instead of $\mathbb N$. With this, it makes sense it is a typo but It still feels weird that it is repeated throughout the paper.