I have been learning about covering maps, and I am having trouble proving something which my intuition is telling me should be true.
For this question, a covering map is a continuous, surjective map $q : E \to X$ where $E$ is locally path connected and connected, and every point in $X$ has an evenly covered open neighbourhood.
Assume that $A \subseteq X$ is simply connected. Then $q^{-1}(A)$ is a disjoint union of sets (just take connected components in the subspace topology). Let $W$ be one of these connected components.
Question: Does $q$ restrict from a homeomorphism from $W$ onto $A$?