Prove that this covering map is a homeomorphism

Prove that this covering map is a homeomorphism

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?