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 $X = \operatorname{St}(K,\mathscr{U})$.
$St(K, \mathscr{U})=\bigcup\{U\in \mathscr{U}: U \cap K \neq \emptyset\}$
$ St^{n+1} (K, \mathscr{U}) = \bigcup \{ U \in \mathscr{U} : U \cap St^{n}(K, \mathscr{U}) \neq \emptyset \}$.
A topological space $X$ is said to be n-star compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a finite subset $\mathscr{V}$ of $\mathscr{U}$ such that $X = \operatorname{St}^{n}(\bigcup \mathscr{V} ,\mathscr{U})$.
1: I think that star compact is the same 1- star compact. Is it right?
2: Is star compact, $n$-star compact?
3: Is there an example that show closed subset in star compact space is not star compact?
