# star compact space

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.

• 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？ – YuiTo Cheng May 1 at 12:02
• 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. – Alex Ravsky Jun 17 at 6:05