Theorem 18.1:

Let $X$ and $Y$ be connected locally path connected spaces, $p : (\tilde{X}, \tilde{x}_0) \longrightarrow (X, x_0)$ is a covering projection and $f: (Y, y_0) \longrightarrow (X, x_0)$ is a continuous function. A lift $\tilde{f} : Y \longrightarrow \tilde X$ satisfying $\tilde{f}(y_0) = {\tilde x}_0$ exists if and only if

$${f}_{*}(\pi_1(Y, y_0)) \subset p_{*}(\pi_1(\tilde{X}, {\tilde x}_0)). \eqno(18.1)$$

In particular, if $Y$ is simply connected, that is if $\pi_1(Y, y_0)$ is trivial, then (18.1) holds and the lift ${\tilde f} :Y\longrightarrow {\tilde X}$ satisfying ${\tilde f}(y_0) = {\tilde x}_0$ exists.