Let $p\colon E\to X$ be a covering map and $f\colon Z\to X$ a topological embedding. Suppose $F\colon Z\to E$ is a lift of $f$. Is $F$ still an embedding? What if I assume that $Z$ is connected and locally connected?
It seems that this is a trivial question. Somehow, I just couldn't answer it. Any help?
Let $g\colon f(Z) \to Z$ denote the inverse of $f$. By definition of an embedding, $g$ is continuous. Now
$$g \circ p \colon F(Z) \to Z$$
is clearly a continuous map, and $g \circ p$ is the inverse of $F$. Hence $F$ is an embedding.