TITLE: topological KC - space
QUESTION [5 upvotes]: A topological space X is KC – space if every compact subsets are closed.
question:
Does a KC - space contains a minimal KC topology?

REPLY [4 votes]: The answer is no; Bill Fleissner constructed a counterexample in ‘A $T_B$ space which is not Katetov $T_B$’, which is freely available here. (He uses the term $T_B$ space for your KC-space.)