Spaces where all compact subsets are closed All compact subsets of a Hausdorff space are closed and there are T$_1$ spaces (also T$_1$ sober spaces) with non-closed compact subspaces. So I looking for something in between.
Is there a characterization of the class of spaces where all compact subsets are closed? Or at least, is there a name for them? 

(This is more an overlong comment:)
In a constructive reading, "all compact subsets are closed" becomes "the identity is a well-defined continuous injection from the space of compact subsets to the space of closed subsets". The latter property is equivalent to being Hausdorff (Proposition 15 in http://arxiv.org/abs/1204.3763).
Thus, for any KC-space that is not Hausdorff, proving that it is indeed KC will require some non-uniform argument. 
