This is a prelim problem which I did not get correctly.
Let $p: X\to Y$ be a closed map. Suppose that $p^{-1}(\{y\})$ is compact for all $y\in Y$. Show that if $Y$ is compact then $X$ is compact.
In the prelim $p$ is not described as continuous so I assumed that it is not necessarily continuous. However I only get to the point where $p(X)$ is compact and I couldn't see how to use the assumption that $p^{-1}(\{y\})$ is compact for all $y$. Any help will be appreciated.
Suppose for the sake of contradiction that $X$ is not compact. We may thus take a net $(x_\alpha){\alpha \in I}$ in $X$ with no convergent subnet. Then ${x\alpha}{\alpha \in I}$ is a closed set in $X$, so ${p(x\alpha)}{\alpha \in I}$ is a closed set in $Y$. But since $Y$ is compact, it has a convergent subnet. So there is some $\gamma \in I$ and subnet $(\alpha\beta)$ with $p(x_{\alpha_\beta}) \to p(x_\gamma)$. Now use that $p^{-1}(p(x_\gamma))$ is compact...
– mathworker21 May 13 '19 at 23:47