Question about two simple problems on covering spaces Here are two problems that look trivial, but I could not prove.
i) If $p:E \to B$ and $j:B \to Z$ are covering maps, and $j$ is such that the preimages of points are finite sets, then the composite is a covering map.
I suppose that for this, the neighborhood $U$ that will be eventually covered by the composite will be the same that is eventually covered by $j$, but I can´t prove that the preimage can be written as a disjoint union of open sets homeomorphic to $U$.
ii) For this I have no idea what to do, but if I prove that it´s injective, I'm done. Let $p:E \to B$ be a covering map, with $E$ path connected and $B$ simply connected; prove that $p$ is a homeomorphism.

Lets call an open neighborhood $U$ of a point $y$ principal (wrt. a covering projection $p: X \to Y$), if it's pre image $p^{-1}(U)$ is a disjoint union of open sets, which are mapped homeomorphically onto $U$ by $p$.
By definition a covering projection is a surjection $p: X \to Y$, such that every point has a principal neighborhood. It is easy to see, that if $U$ is a principal neighborhood of a point $y$, then any open neighborhood $U'$ of $y$ with $U' \subset U$ is again principal.
i) Let $p: X \to Y$ and $q: Y \to Z$ be covering projections, where $q^{-1}(\{z\})$ is finite for every $z \in Z$. Let $z \in Z$ and $U$ a principal neighborhood of $z$. For every point $y \in q^{-1}(\{z\})$ choose a principal neighborhood $V_y$. We can assume that $V_y$ is a subset of the component of $q^{-1}(U)$ corresponding to $y$, possibly replacing $V_y$ with its intersection with that component.
Now let $$U' = \bigcap_{y \in q^{-1}(\{z\})}q(V_y),$$ then $U'$ is an open (being the intersection of finitely many open subsets) neighborhood of $z$. It should be easy to verify that $U'$ is principal.
ii) Let $e,e' \in E$ with $p(e) = p(e')$ and $\gamma: I \to E$ be a path from $e$ to $e'$. Now $p \circ \gamma$ is a closed path, and therefore nullhomotopic. Lifting such a homotopy shows that $e=e'$.
