Let $A\subset X$ and $p: E \to X$ be a covering space. Assume that $A,X,E$ are all locally path connected, path connected, Hausdorff. Suppose that $p^{-1}(A)$ is path connected.
I want to show that $p^{-1}(A)$ is locally path connected. It is a preimage of a locally path connected space, also a subspace of a locally path connected, but I think these do not help. To check directly if $x\in U \subset p^{-1}(A)$, then $p(x) \in p(U) \subset A$ but $p(U)$ need not be an open set.
How can I prove it?