If not, then there are two collections of disjoint open sets $\{A_{\alpha}\}$ and $\{B_{\beta}\}$ in $E$ such that the union of each collection is $p^{-1}(U)$ , and two open sets $A=A_{\alpha}$ for some $\alpha$ and $B=B_{\beta}$ for some $\beta$ such that $A\neq B$ , $A\cap B\neq\emptyset$ , and both $p|_{A}$ and $p|_{B}$ are homeomorphisms of $A$ and $B$ , respectively, with $U$ . Without loss of generality, $A-B\neq\emptyset$ . $A,B$ are both open and closed in $p^{-1}(U)$ , so that $A-B=A\cup B-B$ is open and closed in $p^{-1}(U)$ , and its image under the homeomorphism $p|_{A}$ is a non-empty proper both open and closed subset of $U$ , contradicting the fact that $U$ is connected.