  Fredholm Lie groupoids were introduced by Carvalho, Nistor and Qiao as a tool
for the study of partial differential equations on open manifolds. This article
extends the definition to the setting of locally compact groupoids and proves
that \enquote{the Fredholm property is local}. Let $\mathcal{G}
\rightrightarrows X$ be a topological groupoid and $(U_i)_{i\in I}$ be an open
cover of $X$. We show that $\mathcal{G}$ is a Fredholm groupoid if, and only
if, its reductions $\mathcal{G}^{U_i}_{U_i}$ are Fredholm groupoids for all $i
\in I$. We exploit this criterion to show that many groupoids encountered in
practical applications are Fredholm. As an important intermediate result, we
use an induction argument to show that the primitive spectrum of
$C^*(\mathcal{G})$ can be written as the union of the primitive spectra of all
$C^*(\mathcal{G}|_{U_i})$, for $i \in I$.
