# Is a proper quotient map closed ?

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.

-
By your definition, a quotient map does not have to be onto. Is this deliberate? (If so, the answer to your question is “no”.) –  Harald Hanche-Olsen Mar 19 '10 at 13:19
Never mind that. See my answer below. –  Harald Hanche-Olsen Mar 19 '10 at 13:29