Path lifting property

Remarks

For existence, cover $X$ by evenly covered open sets. By the Lebesgue number lemma, there is a subdivision $0 = s_0 < s_1 < \cdots < s_k = 1$ of the interval with each $\gamma([s_{j-1}, s_j])$ in one evenly covered set $U_j$. The lift is built step by step: on $[s_{j-1}, s_j]$ it is $(p|_{V})^{-1} \circ \gamma$, where $V$ is the sheet over $U_j$ that contains the point already reached. Uniqueness holds because the set where two lifts agree is open and closed in the connected interval $I$, and it contains $0$. Example: for $p(t) = e^{2\pi i t}$ from $\R$ to $S^1$ and $\gamma(s) = e^{2\pi i n s}$, the lift starting at $0$ is $\tilde\gamma(s) = ns$, which ends at $n$. The covering hypothesis is needed: for the local homeomorphism $q = p|_{(0, 2)} \colon (0, 2) \to S^1$, the loop $\gamma(s) = e^{-2\pi i s}$ with $e_0 = 1$ has no lift. A lift in $(0, 2)$ is also a lift in $\R$, so by uniqueness it would be $s \mapsto 1 - s$, whose value at $s = 1$ is $0 \notin (0, 2)$.

Going deeper

For a student

The path lifting property, together with its extension to homotopies, is the main tool of covering space theory. It shows that $p_* \colon \pi_1(E, e_0) \to \pi_1(X, p(e_0))$ is injective, and it computes the fundamental group of the circle through the covering $\R \to S^1$. The endpoint $\tilde\gamma(1)$ of the lift of a loop defines an action of $\pi_1(X, x_0)$ on the fibre $p^{-1}(x_0)$, which leads to the classification of covering spaces.