Homotopy lifting for covering maps

Remarks

For each $x \in X$, cover $\{x\} \times [0, 1]$ by sets of the form $N \times [t_i, t_{i+1}]$ that $H$ maps into evenly covered open sets of $B$. On each such piece the lift is forced: it is the composite with the inverse of $p$ on the sheet that contains the lift already built. Uniqueness of lifts on the connected sets $\{x\} \times [0, 1]$ makes these local lifts agree, so they glue to a continuous $\tilde H$. When $X$ is a point, the statement is the path lifting property: each path in $B$ lifts uniquely once the start point of the lift is chosen. Applied to $X = [0, 1]$ and a homotopy of loops rel endpoints, it shows that $p_* \colon \pi_1(E, e_0) \to \pi_1(B, p(e_0))$ is injective, where $\pi_1$ is the fundamental group. For example, take $p \colon \R \to S^1$, $p(t) = (\cos 2\pi t, \sin 2\pi t)$, and a loop at $(1, 0)$ whose lift runs from $0$ to $n \ne 0$. This loop is not null-homotopic, because a null-homotopy would lift to a homotopy rel endpoints in $\R$.

Going deeper

For a student

This property is the engine of covering space theory. It gives the injectivity of $p_*$ on fundamental groups, the computation $\pi_1(S^1) \cong \Z$, and the lifting criterion behind the classification of covering spaces. It also says that a covering map is a fibration with discrete fibers, so the long exact sequence of homotopy groups applies and gives $\pi_n(E) \cong \pi_n(B)$ for $n \ge 2$.