Let $p \colon E \to X$ be a covering map. Let $s \colon X \to E$ be continuous. If $p \circ s = \operatorname{id}_{X}$, show that $p$ is a homeomorphism.
We know that $p$ is a continuous surjection. Since all covering maps are open, we just need to show that $p$ is an injection. How is this done?
Let $x,\, y \in E$ with $p(x) = p(y)$. Since $E$ is path-connected (follows from connected and locally path-connected), there is a path $\gamma$ connecting $x$ and $y$.
$\beta := p\circ \gamma$ is a closed path in $X$. $\gamma$ is a lift of $\beta$, and $s\circ \beta$ is also a lift of $\beta$. By the uniqueness of lifts, $\gamma = s\circ \beta$.
But $s\circ \beta$ is of course a closed path, hence $y = x$.