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? 

According my acknowledge, it hasn't a characterization of this class of spaces, I believe that the reason is that this class of spaces is not Hausdorff and many people don't care it since we always study the classes of spaces at least Hausdorff.
The following paragraphes may be useful for you, which is copied from Page 221 Encyclopedia of General Topology: 
A result taught in a first course in topology is that a compact
subspace of a Hausdorff space is closed. A Hausdorff
space with the property of being closed in every Hausdorff
space containing it as a subspace is called H-closed (short
for Hausdorff-closed).
H-closed spaces were introduced in
1924 by Alexandroff and Urysohn. They produced an example
of an H-closed space that is not compact, showed that
a regular H-closed space is compact, characterized a Hausdorff
space as H-closed precisely when every open cover has
a finite subfamily whose union is dense, and posed the question
of which Hausdorff spaces can be densely embedded in
an H-closed space.
