When explaining the degree of a covering map, my lecture notes say:
Let $p: \bar Y \to Y$ be a covering map. Let $V \subset Y$ be a well-covered set (in the sense that its preimage consists of a disjoint union of open sets each of which is mapped homeomorphically to $V$ by $p$).
Let $y \in V$. For each of the open sets $U_\alpha$ making up $P^{-1}(V)$ as in the definition of covering map, $U_\alpha \cap p^{-1}(y)$ consists of a single point.
Why does each $U_\alpha$ have to contain a $p^{-1}(y)$? Can it not be the case where a $U_\alpha$ does not contain a $p^{-1}(y)$?