Question

I am trying to produce closed quotient maps, as they allow a good way of creating saturated open sets (as in this question).

A map $f:X\rightarrow Y$is called proper, iff preimages of compact sets are compact. It is called quotient map, iff a subset $V\subset Y$ is open, if and only if its preimage $f^{-1}(V)$ is open. And it is called closed, iff it maps closed sets to closed sets.

So the question is, whether a proper quotient map is already closed.

Note that, I am particular interested in the world of non-Hausdorff spaces.

Answer 1

No. Let X={1,2,3} and Y={1,2}. Let f map 1 to 1, 2 and 3 to 2. Let the topology on X be {∅,{2},{1,2},{2,3},{1,2,3}} and that on Y be {∅,{2},{1,2}}. f maps the closed set {3} onto the non-closed set {2}.

Answer 2

Answer removed, was same as Harald's