Theorem 17.4:

For a covering projection $p: {\tilde X}\longrightarrow X$ with path connected $X$ and ${\tilde X}$ the following are equivalent:

(i)

The subgroup $p_*(\pi_1({\tilde X}, {\tilde x}_0))$ is a normal subgroup of $\pi_1(X, x_0)$, where $p({\tilde x}_0) = x_0$.

(ii)

For any loop in $X$ based at $x_0$, either all its lifts are closed loops or none of the lifts is closed.