From C. Godbillon's Eléments de Topologie Algébrique, chapter VII (Covering Spaces), section 1 (Local homeomorphisms). We have the following problem :
Let $p : E \to I$ be a local homeomorphism from a connected, Hausdorff space $E$ to the unit interval $I=[0,1]$. If $p$ is surjective, then it is a homeomorphism.
It is enough to show that $p$ is injective; however it is not entirely clear to me how to prove this.
Moreover I am interested in the "optimality" of this result : what properties of the unit interval (because this is generally not true if one replaces $I$ by some other space) allow this to be true, and can we deduce from this a more general statement ? Any help would be appreciated.
(For additional context, I think that one could use this result to show that every covering space on $I$ (and therefore $I^n$) is trivial).