# star-compact topological 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.

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 ?

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.