Proposition 8

Let X and Y be non-empty topological spaces and let $f$ be a continuous mapping of X into Y. Assume that the space X is locally arcwise connected, that the space Y is connected, and that the mapping $f$ is strict and surjective. Then the groupoid Coeg($f$) is transitive.

Let Γ be the quiver whose set of vertices is X and whose set of arrows is the sum set of Fl($\varpi (X)$) and $X\times_YX$, the origin and target mappings being those of $\varpi (X)$ on Fl($\varpi (X)$) and the mappings $p_1$ and $p_2$ on $X\times_YX$. By the definition of the framework of the pair $(\varpi$(pr$_1$)$, \varpi$(pr$_2$)) (II, p. 185, Def. 3) and Remark 2 of II, p. 200, the set of orbits of Coeg($f$) is identified with the set of connected components of the quiver Γ. Since the space X is non-empty, it is enough to prove that the graph Γ is connected.

The connected components of Γ are saturated with respect to the equivalence relation “there exists a path joining $x$ to $x'$”, and therefore are open in X, since X is locally arcwise connected. They are then also closed. They are also saturated with respect to the equivalence relation R defined by $f$. By hypothesis, the mapping $f$ induces a homeomorphism of $X/R$ onto Y, so that the image under $f$ of every connected component of Γ is an open and closed subset of Y, and therefore is equal to Y since the space Y is assumed connected.

Let C be a connected component of Γ, and let $x$ be a point of X. From the preceding, there exists a point $x'\in C$ such that $f(x') =f(x)$. By definition of the quiver Γ, we have $x'\in C$. Thus C = X and the quiver Γ is therefore connected.