Corollary 3

Let E, B and Z be topological spaces, let $p: E\rightarrow B$ be an étale and separated mapping, and let $f: Z\rightarrow B$ be a continuous mapping. For $i= 1$, 2, let $U_i$ be an open subset ( resp. closed subset) of Z and $g_i: U_i\rightarrow E$ a continuous lifting of $f$ defined on $U_i$. Suppose that the intersection $U_1\cap U_2$ is connected and that there exists a point $z$ of $U_1\cap U_2$ such that $g_1(z) =g_2(z)$. Then there exists a continuous lifting of $f$ defined on $U_1\cup U_2$ which extends $g_1$ and $g_2$.

By Corollary 1, the restrictions of $g_1$ and $g_2$ to $U_1\cap U_2$ are equal. The mapping $g: U_1\cup U_2\rightarrow E$ defined by $g(z) =g_i(z)$ for $z\in U_i(i= 1$, 2) is a continuous lifting of $f$ defined on $U_1\cup U_2$ (TG, I, p. 19, Prop. 4).

In the preceding results, the particular case where Z = B and $f=$ Id$_B$ is important: the continuous liftings of $f$ are then the continuous sections of $p$.