Inverse image of a point under a continuous surjective closed map :

Question

Let $f:X\rightarrow Y$ be a continuous surjective closed map and $X$ is a normal space. Let there exist an open set $U\subset X$ such that $f^{-1}\{ y \}\subset U$ then show that there exist an open set $W\subset Y$ and $y\in W$ such that $f^{-1}(W)\subset U$.

Unable to get choice of $W$ . $W=Y-f(U^c)$ is an open set that contains $y$ but $f^{-1}(W)$ may not be contained in $U$. So how to prove this?

Answer 1

Your choice of $W$ is fine. If $z \in f^{-1}(W)$ then $f(z) \in W$ which implies $f(z) \notin f(U^{c})$. This implies $ z\notin U^{c}$ so $z \in U$.