I have encountered, in different circumstances, the following two slightly different categories:
The full category of $\mathsf{Top}$ consisting of all objects that are:
a) topological spaces such that all compact subspaces are closed and that any subset is closed iff its preimage of every embedding of a compact subspace is closed (denotes by $\mathsf A$);
or
b) topological spaces such that all images of Hausdorff compact spaces are closed and that any subset is closed iff its preimage of every continuous map from a Hausdorff compact space is closed (denotes by $\mathsf B$).
I find that $\mathsf B$ is contained in $\mathsf A$, but I have no idea if $\mathsf B$ is properly contained in $\mathsf A$. Does anyone know the answer?
Besides, if $\mathsf A$ and $\mathsf B$ are distinct. I'm also wondering what's the advantages and disadvantages of using $\mathsf A$ instead of $\mathsf B$, and conversely?