Proposition 5

Let X be a topological space and let $(A_i)_{i\in I}$ be an increasing family of subsets of X, indexed by a filtered ordered set I, such that every quasi-compact subset of X is contained in one of the $A_i$. The canonical groupoid morphism

$\rho :$ lim$\longrightarrow_{i\in I}\varpi (A_i)\rightarrow \varpi (X)$,

induced by the canonical injections of $A_i$ into X, is an isomorphism.

If $i\leqslant j$, let us denote by $\rho_{j,i}$ the groupoid morphism $\varpi (A_i)\rightarrow \varpi (A_j)$ induced by the injection of $A_i$ into $A_j$. Since the mapping induced by $\rho_{j,i}$ by passing to the vertices is the injection $A_i\rightarrow A_j$ and since the $A_i$ cover X, the mapping induced by $\rho$ by passing to the vertices is bijective.

Let $a$ and $b$ be points of X and let $c$ be a path joining $a$ to $b$ in X. The image of $c$ is a quasi-compact subset of X (TG, I, p. 62, Theorem 2), since $\mathbf{I}$ is compact. There therefore exists an element $i\in I$ such that the image of $c$ is contained in $A_i$. Consequently, the mapping induced by $\rho$ by passing to the sets of arrows is surjective.

Let $i\in I$, let $a$ and $b$ be points of $A_i$, let $c,c'$ be paths joining $a$ to $b$ in $A_i$; let $h$ be a strict homotopy joining $c$ to $c'$ in X. Since $\mathbf{I}\times \mathbf{I}$ is compact, $h(\mathbf{I}\times \mathbf{I})$ is a quasi-compact subset of X (loc. cit.) and there exists an element $i\in I$ such that the image of $h$ is contained in $A_i$. The paths $c$ and $c'$ are strictly homotopic in $A_i$; fortiori, the path classes $[c]$ and $[c']$ have the same image in lim$\longrightarrow \varpi (A_i)$. Consequently, $\rho$ is injective.