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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.

So,

(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?

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  • $\begingroup$ You already know the result of Bella & Costantini that every minimal $KC$ space is compact, so the answer to (1) is no. $\endgroup$ – Brian M. Scott Aug 29 '13 at 18:58
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You already know the result of Bella & Costantini that every minimal $KC$ space is compact, so the answer to (1) is no. Section $2$ of Chiara Baldovino & Camillo Costantini, ‘On some questions about $KC$ and related spaces’, Topology and its Applications $156$ $(2009)$, No. $17$, $2692$-$2703$, gives the construction of a compact $KC$ space in which every non-empty open set is dense. Example $3.1$ in the same paper is an example of a minimal Hausdorff space that is not a $k$-space.

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