Corollary

Let X and B be topological spaces and let $p: X\rightarrow B$ be a continuous, open, proper and separated mapping whose fibres are totally disconnected. Let I be an interval of $\mathbf{R}$, let $f: I\rightarrow B$ be a continuous mapping, let $a$ be a point of I and let $x$ be a point of X such that $f(a) =p(x)$. There exists a continuous mapping $g: I\rightarrow X$ such that $p\circ g=f$ and $g(a) =x$.

Posons $X'= I\times_BX$ and denote by $p': X'\rightarrow$ I and $f': X'\rightarrow$ X the canonical projections. The mapping $p'$ is continuous, open, proper and separated (I, p. 17, prop. 8 and p. 27, prop. 4). Since the space I is separated, the space $X'$ is separated (I, p. 26, remark 3). The fibres of $p'$ are totally disconnected ( I, p. 10, corollary, a)). According to Theorem 3, there exists a continuous section $s'$ of $p'$ which takes in $a$ the value $(a, x)$. The mapping $g=f'\circ s'$ of I into X is continuous, we have $p\circ g=p\circ f'\circ s'=f\circ p'\circ s'=f$ and $g(a) =x$, whence the corollary.