Theorem 3

Let I be an interval of $\mathbf{R}$ and let X be a nonempty and separated topological space. Let $p: X\rightarrow I$ be a continuous, open and proper mapping whose fibres are totally disconnected (TG, I, p. 83). The mapping $p$ is surjective. For every point $x$ of X, it possesses a continuous section $s$ such that $s(p(x)) =x$.

The set $p(X)$ is an open, closed (TG, I, p. 72, prop. 1), nonempty subset of I. Since I is connected, one has $p(X) = I$; the mapping $p$ is therefore surjective.

For every pair $(a, b)\in I\times I$ such that $a\leqslant b$, let $F_{a,b}$ denote the set of pairs $(y, z)\in \overset{-1}{p}(a)\times \overset{-1}{p}(b)$ such that $y$ and $z$ belong to one and the same connected component of $\overset{-1}{p}([a, b])$.