We call a map proper if the inverse image of quasi-compact sets are quasi-compact sets.(Add the Hausforff axiom if convient) Is there example for such a map not be a closed map?
(If the spaces are locally compact Hausdorff, any proper map would be closed.)
It’s not necessarily true without some separation axioms. Let $\tau_D$ be the discrete topology on $X=\{0,1\}$, and let $\tau_I$ be the indiscrete topology on $X$. Then the identity map from $\langle X,\tau_D\rangle$ to $\langle X,\tau_I\rangle$ is continuous and proper but not closed. You can replace replace $\tau_I$ by the Sierpiński topology, which is $T_0$, and still have an example.
