Theorem 16.1 (path lifting lemma):

Let $p: {\tilde X}\longrightarrow X$ be a covering projection and $\gamma :[0, 1] \longrightarrow X$ be a path such that for some $x_0 \in X$ and ${\tilde x}_0 \in {\tilde X}$,

$$\gamma(0) = x_0 = p({\tilde x}_0). \eqno(16.1)$$

Then there exists a unique path $\tilde{\gamma}: [0, 1] \longrightarrow {\tilde X}$ such that

$$p\circ \tilde{\gamma} = \gamma,\quad {\tilde\gamma}(0) = \tilde{x}_0\eqno(16.2)$$

Thus each path in $X$ lifts to a unique path in ${\tilde X}$ with a prescribed initial point in $p^{-1}(\gamma(0))$. in