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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 $X = \operatorname{St}(K,\mathscr{U})$.

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

‎‎‎$ ‎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?‎‎

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  • $\begingroup$ 1 depends on your definition of $1$-star compact. $\operatorname{st}^n(K, \mathcal{U}) \subseteq \operatorname{st}^{n+1}(K, \mathcal{U})$ for all $n$. That answers 2. $\endgroup$ – Henno Brandsma Jul 14 at 11:29
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    $\begingroup$ Define $n$-star compact ?? $\endgroup$ – Henno Brandsma Jul 14 at 15:03
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    $\begingroup$ I could not find a counterexample nor a theorem regarding 3 in a quick online literature survey.. $\endgroup$ – Henno Brandsma Jul 14 at 15:19
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    $\begingroup$ You did neither define $1$-star compact nor $n$-star compact, but certainly you have $St^1(K, \mathscr{U})= St(K, \mathscr{U})$ and say that $X$ is $n$-star compact if for every open cover $\mathscr{U}$ of $X$ there is a compact subspace $K$ of $X$ such that $X = \operatorname{St}^n(K,\mathscr{U})$. Moreover, you may define $St^0(K, \mathscr{U})= K$. Then the definition $‎St‎^{n+1}‎‎ (K, \mathscr{U}‎) =‎ ‎\bigcup ‎ \{ U‎ ‎\in‎ \mathscr{‎U} : U ‎\cap St‎^{n}‎‎‎(K, \mathscr U ‎) ‎\ne ‎\emptyset \}‎‎‎ ‎$‎ is valid for all $n \ge 0$. $\endgroup$ – Paul Frost Jul 14 at 17:27
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  1. Yes, see Remark 2.1 in the paper “On $\mathcal K$-Starcompact Spaces” by Yan-Kui Song (Bull. Malays. Math. Sci. Soc. (2) 30:1 (2007), 59–64).
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