A typo in my lecture notes made me wonder if these definitions are actually equivalent. Let $\tilde X \xrightarrow{p} X$ be continuous and surjective.
- (actual definition of covering spaces) For any $x \in X$ we find $U$ a neighborhood of $x$ such that $p^{-1}(U) = \bigsqcup\limits_{i \in I} U_i$ ($U_i \subseteq \tilde X$ open and pairwise disjoint) such that $U_i$ is homeomorphic to $U$ via $p$.
- Same as before, only instead of $U_i$ homeomorphic to $U$ via $p$, we just require that $p|_{U_i}$ be injective and $U_i$ be homeomorphic to $U_j$ for any $i, j \in I$.
My intuition says these two are very likely not equivalent (of course, the first implies the second). However, is there an example of a $p$ which satisfies (2) but not (1)?