For the purposes of my question, a continuous map $f : X \to Y$ is proper if it is closed and the preimage of every compact subspace of $Y$ is a compact subspace of $X$.
Say a continuous map $f : X \to Y$ is semiproper if, for every continuous map $y : T \to Y$ where $T$ is compact, the space $T \times_Y X = \{ (t, x) \in T \times X : y (t) = f (x) \}$ is compact.
It is a fact that a closed map is proper if and only if it is semiproper.
Question. Are semiproper maps always closed?
If $Y$ is a compactly generated Hausdorff space, then it is easy to check that every semiproper map $f : X \to Y$ is closed – indeed, we only need the defining property for subspace inclusions $y : T \to Y$. On the other hand, if we weaken the definition by restricting to subspace inclusions $y : T \to Y$, then there are easy counterexamples.
That leaves non-(compactly generated Hausdorff) spaces. Perhaps there is a counterexample there?