Number of partition into pieces (covering map)

Let $U\subset B$ be evenly covered by $p:E\rightarrow B$ and $\{V_\alpha\}$ be the partition of $p^{-1}(U)$ into pieces. Number of elements of $\{V_\alpha\}$ is equal to the number of elements of $p^{-1}(b)$ for all $b\in U$.

Because each of $p|_{V_\alpha}$ is a homeomorphism, exactly one element of $p(b)$ exists in $V_\alpha$. No more, no less.