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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.

1: Let $(X, \tau_1)$ be a star-compact topological space and $\tau_2 \subset \tau_1$.Is $(X , \tau_2)$ a star-compact topological space?

we know that Hausdorff compact space is minimal Hausdorff or maximal compact.

2:Is there relation between (minimal)Hausdorff and maximal star-compact? I mean, is it possible to deduce from (minimal) Hausdorff a star maximal compact space Or vice versa? If there is no relation, what conditions are needed to let Hausdorff's space become a maximal star compact ?

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The first question seems rather easy:

Let $\mathcal{U}$ be a $\tau_2$-open cover of $X$, which is also a $\tau_1$-open cover of $X$ (as $\tau_2 \subseteq \tau_1$). So there is a $\tau_1$-compact $K$ with $\operatorname{St}(K, \mathcal{U}) = X$. But $K$ is also $\tau_2$-compact so we are done and the same $K$ shows that $(X, \tau_2)$ is star compact.

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