Corollary

Let X and Y be contractible topological spaces and let $f: X\rightarrow Y$ be a continuous and relatively connected mapping. Suppose that every point of Y possesses a neighbourhood V whose inverse

image $\overset{-1}{f}(V)$ is simply connected. Then, for every point $x$ of X, the homomorphism $\pi_1(f, x):\pi_1(X, x)\rightarrow \pi_1(Y, f(x))$ is bijective.

One may suppose the spaces X and Y connected (IV, p. 353, prop. 4). By corollary 3 (IV, p. 357), the homomorphism $\pi_1(f, x)$ is surjective. Proposition 6 and proposition 1 of IV, p. 351 imply that it is injective.