We know that,
Let $X$ and $Y$ be two topological spaces and $p:X\to Y$ be a $\color{red}{\text{closed}}$ continuous surjective map where $Y$ is compact. Let for all $y\in Y$, $p^{-1}(\{y\})$ is also compact. Prove that $X$ is compact.
The proof of this result can be found here.
However, I was wondering if the following theorem also holds,
Let $X$ and $Y$ be two topological spaces and $p:X\to Y$ be a $\color{red}{\text{open}}$ continuous surjective map where $Y$ is compact. Let for all $y\in Y$, $p^{-1}(\{y\})$ is also compact. Then $X$ is compact.
In case it is false, can someone give an counterexample to the above proposition? In can it is true, can someone give me an hint for the proof?