Suppose $X=U\cup V$ where $U$ and $V$ are open. Then, each $p^{-1}(\{y\})$ being connected must lie within either $U$ or $V$ . Therefore, $U$ and $V$ are saturated. Then, by a remark after the definition on page 137, $p(U)$ and $p(V)$ are open. Moreover, they are disjoint (being images of disjoint saturated sets), and, since $p$ is surjective, $p(U)\cup p(V)=Y$ . Since $Y$ is connected, we conclude that one of the sets $U$ or $V$ must be empty.