Proposition 11

Let E, B and Z be topological spaces, $p: E\rightarrow B$ and $f: Z\rightarrow B$ continuous mappings. Let $g$ and $g'$ be continuous liftings of $f$ to E. Let W be the set of points of Z where $g$ and $g'$ coincide.

a) If the mapping $p$ is separated, W is closed.

b) If the mapping $p$ is étale, W is open.

Let us denote by $h$ the continuous mapping $(g, g'): Z\rightarrow E\times E$. The image of $h$ is contained in $E\times_BE$ since $p\circ g=f=p\circ g'$ and the set W of points of Z where $g$ and $g'$ coincide is the inverse image under $h$ of the diagonal $\Delta_E$.

If the mapping $p$ is étale, the diagonal $\Delta_E$ is open in $E\times_BE$ (I, p. 31, prop. 7), therefore the set W is open in Z.

If the mapping $p$ is separated, the diagonal $\Delta_E$ is closed in $E\times_BE$ (I, p. 25, def. 1) and the set W is closed in Z.