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A topological space $X$ is said to be star compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a compact subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$.

$St(K, \mathscr{U})=\cup\{u\in \mathscr{U}: u \cap K \neq \emptyset\}$

If P is a topological property, then a space $(X, \tau)$ is said to be minimal $P$ (respectively, maximal) if $(X, \tau)$ has property $P$ but no topology on $X$ which is strictly smaller (respectively, strictly larger ) than τ has P.

A topological space is called KC space if every compact subset is closed.

So, My questions:

1:I know that every $KC$- minimal space is compact , but is every star compact, $KC$- minimal space?

2: Is there relation between star compact and Hausdorff space?

Thanks.

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    $\begingroup$ I don't think every star compact space is countably compact, not to mention compact, according to Brian's answer on How far is being star compact from being countably compact? $\endgroup$ – YuiTo Cheng May 1 at 12:02
  • $\begingroup$ I expect that there is no non-artificial relations between star compact and Hausdorff spaces. The standard space of reals is Haudorff but not star compact, where a two-point set endowed with the antidiscrete topology is a (star) compact non-Hausdorff space. $\endgroup$ – Alex Ravsky Jun 17 at 6:05

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