Corollary 1

Let B be a topological space and let E be a covering of B; let us denote by $p$ its projection. For every pair $(x, y)$ of points of E, the mapping $\varpi_{x,y}(p):\varpi_{x,y}(E)\rightarrow \varpi_{p(x),p(y)}(B)$ is injective. In particular, for every point $x$ of E, the homomorphism $\pi_1(p, x):\pi_1(E, x)\rightarrow \pi_1(B, p(x))$ is injective.