$f:X→Y$ is a closed map. Suppose that the inverse image of each point in $Y$ is a compact subset of $X.$ Show that $f^{-1}(K)$ is compact in $X$ whenever $K$ is compact in $Y.$
This is my homework and I understand $Y$ is a $T_1$ space. But what should I do next?
A map $f:X\to Y$ which is continuous and closed and has the property that all fibers (the preimages of points) are compact is called perfect. A map is called proper if the preimages of compact sets are compact. So you want to show that each perfect map is proper.
Note that for every $B\subset Y$, the restriction $f':f^{-1}(B)\to B$ is again proper. In particular, if $K\subseteq Y$ is compact, then $f':f^{-1}(K)\to K$ is a perfect map with a compact target space. For that case, you can find a proof at Perfect Map $p:\ X\to Y$, $Y$ compact implies $X$ compact which also shows that the continuity is not needed.
