This is some attempt to generalise the classical result for a compact Hausdorff space $D$, if $E\subseteq D\times D$ is closed, then the quotient map $p:D\to D/{E}$ is a closed equivalence relation. Below $D$ is not Hausdorff anymore (but (quasi)-compact and $D\times D$ is considered with topology that is finer than product topology in terms of closed sets).
Let $D$ be a Noetherian topological space. Let $E\subseteq D\times D$ be a closed equivalence relation. Note that the topology on $D\times D$ is not necessarily the product topology, but it is a Noetherian topology which is finer than the product topology with respect to closed sets (e.g. as for algebraic varieties, I am aware that this is a rather vague statement. I am thinking on it to clarify further).
Consider $T := D/{E}$ with the natural quotient topology, and let $p:D\to T$ be the quotient map. Is it true that $p$ is a closed function, i.e. is the image of a closed set under $p$ closed?
Note that $D$ is not necessarily Hausdorff, but it is compact (or quasi-compact in algebraic geometry terminology) in the sense that every open cover has a finite subcover.