Theorem 16.3:

Given a covering projection $p: {\tilde X}\longrightarrow X$, for any $x_0 \in X$ and ${\tilde x}_0 \in {\tilde X}$ the induced group homomorphism

$$p_*:\pi_1({\tilde X}, {\tilde x}_0)\longrightarrow \pi_1(X, x_0)$$

is injective.