TITLE: If a connected open set is evenly covered, then its preimage is uniquely partitioned into slices
QUESTION [7 upvotes]: This is from Topology by Munkres:

Let $p:E \to B$ be a covering map. Suppose $U$ is a open set of $B$ that is evenly covered by $p$. Show that if $U$ is connected, then the partition of $p^{-1}(U)$ into slices is unique.

What I've tried so far
I proved that if $\{V_\alpha\}$ is a slice then each $V_\alpha$ is connected, but I don't know what to do next.

REPLY [5 votes]: I think we could make a simpler argument:
Let $\{U_{\alpha}\}$,$\{V_{\beta}\}$ be partitions of $p^{-1}(U)$.
We know that, if we fix $\alpha_0$,  $\forall x\in U_{\alpha_0}$ $\exists! \beta$ s.t $ x \in V_{\beta}$.
It follows that if $U_{\alpha_0}$ intersects more than two $V_{\beta}$'s we can write $U_{\alpha_0} = \cup_{\beta} [U_{\alpha_0} \cap V_{\beta}]$ a disjoint open partition; a contradiction to $U_{\alpha_0}$ being connected.