I am self studying Topology, and I am having difficulty in proving the following statement.
Let $f$ be a continuous mapping defined on a compact space $X$ onto a Hausdorff space $Y$. Suppose $y\in Y$ and $U$ is an open subset of $X$ that contains $f^{-1}[y]$. Show that there exists an open neighborhood $V$ of $y$ such that $f^{-1}[V] \subset U$.
I know that
- $Y=f[X]$ is compact, and thus, being Hausdorff, it is also normal;
- $f$ is a closed map;
- $f^{-1}[y]$ is closed and compact.
Can you give me some hints on how to use these facts (or other consequences of the hypothesis) to prove the theorem?