If $X$ and $Y$ are topological spaces, $Y$ is compact and $f:X\rightarrow Y$ is a closed map (meaning that $f(A) $ is closed for any closed subset $A$ of $X$) such that $f^{-1} (\{ y\}) =\{x\in X, f(x) =y\} $ is compact $\forall y\in Y$ then $X$ is compact. How do I prove this? Thank you in advance for your help.
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Counterexample: the identity map on $\mathbb{R}$. You need more conditions to prove it. – Hanul Jeon Oct 06 '19 at 11:21
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@Hanul yes sorry, I forgot to say that $Y$ is compact as well. – Tengen Oct 06 '19 at 11:24
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Please edit your question. – J.-E. Pin Oct 06 '19 at 11:26