Proposition 3

Let B be a topological space. Let $(E, p)$ be a covering of B and let $y$ be a point of E. Let X be a simply connected topological space, let $f: X\rightarrow B$ be a continuous mapping and let $x$ be a point of X such that $f(x) =p(y)$. Then there exists a unique continuous lifting $g: X\rightarrow E$ of the mapping $f$ such that $g(x) =y$.

The mappings $g$ sought correspond bijectively (I, p. 9, prop. 3) to the continuous sections $s$ of the covering pr$_1: X\times_BE\rightarrow X$ such that $s(x) = (x, y)$. Such a section exists because the space X is simply connected; it is unique because the space X is connected (I, p. 34, cor. 1 of prop. 11).