Proposition 1

Let A be an unravelable topological space, let B be a topological space locally connected by arcs, and let $f: A\rightarrow B$ be a continuous mapping. Let $a$ be a point of A and $b=f(a)$. Suppose that every covering of A is isomorphic to the inverse image by $f$ of a covering of B. Then the homomorphism $\pi_1(f, a):\pi_1(A, a)\rightarrow$ $\pi_1(B, b)$ is injective.

Since the space A is unravelable, there exists a covering E of A whose fibre $E_a$ is a principal homogeneous $\pi_1(A, a)$-set (IV, p. 342, Theorem 1). By assumption, this operation is obtained by means of the homomorphism $\pi_1(f, a)$ from an operation of $\pi_1(B, b)$ on $\pi_1(A, a)$. This implies the injectivity of the homomorphism $\pi_1(f, a)$.