Corollary 2 (Lifting of paths)

Let $p: E\rightarrow B$ be a covering, let $x$ be a point of E and $a=p(x)$. The mapping $c\mapsto p\circ c$ is a homeomorphism of the space $\Lambda_x(E)$ of paths in E with origin $x$ onto the space $\Lambda_a(B)$ of paths in B with origin $a$.

With the notation of proposition 3, we have $\Lambda_a(B) =o^{-1}_B(a)$ and $\Lambda_x(E) =\overset{-1}{o_{E}}(x)$. The corollary then follows from I, p. 10, cor. of prop. 4.

We shall say that a continuous mapping $p: E\rightarrow B$ satisfies the path-lifting property if, for every path $c:\mathbf{I}\rightarrow$ B and every point $x$ of E lying over $c(0)$, there exists a path $c'$ in E with origin $x$ which lifts $c$. Such a path is then unique if $p$ is étale and separated (I, p. 34, cor. 1 of prop. 11). Let E be a covering and $p$ its projection; the mapping $p$ is étale, separated and has the path-lifting property (corollary 2). For a partial converse, cf. III, p. 315, corollary.