A space $(X,\tau)$ is said to be minimal $KC$ if $(X,\tau)$ is $KC$ but no topology on $X$ which is strictly smaller than $\tau$ is $KC$.
A topological space is called a $k$-space if it has the property that any subset $S$ such that $S\cap K$ is closed for all closed compact $K$ is itself closed.
(1 ) : Can a non-compact minimal Hausdorff space be minimal $KC$?
(2 ) : Every minimal Hausdorff space a $k$-space?
( 3) : Does there exist a compact $KC$-space in which every non-empty open set is dense?