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Definition 1: ‎If ‎‎$‎X‎$‎ is a topological space and ‎‎$‎‎\mathcal{U}‎‎ ‎‎$‎ is a family of subsets of ‎‎$‎X‎$‎, then the star of a subset ‎‎$‎A ‎\subseteq‎ ‎X‎$‎ with respect to ‎‎$‎\mathcal{U}‎‎‎$‎ is the set: ‎‎‎

$‎\textrm{s‎t}(A,\mathcal{U})=\{ U ‎\in ‎‎\mathcal{U}‎‎: U \cap A ‎\neq ‎\emptyset‎ \}‎$‎.

$‎\textrm{s‎t}^{n+1}(A,\mathcal{U})=\textrm{st}(\textrm{s‎t}^{n}(A,\mathcal{U}))=‎‎‎\bigcup\{U‎\in\mathcal{U}‎‎\mid U\cap \textrm{s‎t}^{n}(A,\mathcal{U}) ‎\neq ‎\emptyset‎ \}‎$‎. ‎

Let ‎$‎\mathscr{P}‎$‎ be a topological property. A space ‎‎$‎X‎$‎ is said to be star-‎‎$\mathscr{‎P}‎$ if whenever ‎‎$‎‎\mathcal{U}‎‎$‎‎‎ is an open cover of ‎‎$‎X‎$‎, there is a ‎subspace $‎ A ‎\subseteq‎ ‎X‎$‎ with property ‎‎$‎P‎$‎ such that ‎‎$‎X=‎\textrm{s‎t}(A,‎‎\mathcal{U})‎‎$‎‎, so a space ‎‎$‎X‎$‎ is said to be ‎‎star-compact if whenever ‎‎$‎‎\mathcal{U}‎‎$‎‎‎ is an open cover of ‎‎$‎X‎$‎, there is a compact ‎subspace $‎A‎\subseteq‎ ‎X‎$‎ such that ‎‎$‎X=\textrm{‎s‎t}(A,\mathcal{U})‎‎$. ‎ Theorem 1: Let ‎‎$‎‎\mathcal{U}‎‎‎$ ‎be a‎n ‎open ‎cover, ‎so ‎‎ $\overline{‎\textrm{st}‎^{n}(A,\mathcal{U})‎‎}‎ \subseteq‎‎ \textrm{st}^{n+1}(A, \mathcal{U})$

Definition 2: A Hausdorff topological space ‎‎$‎(X,‎\tau‎)‎‎$‎ is called ‎‎$‎H‎$‎-closed or absolutely closed if it is closed in any Hausdorff space, which contains ‎‎$‎X‎$‎ as a ‎subspace.‎

Theorem 2 :Let ‎‎$‎X‎$‎ be a Hausdorff space. ‎‎$‎‎‎X$ is ‎‎$‎H‎$‎-closed if and only if every open cover ‎‎$\mathcal{C}‎$‎‎‎ of ‎‎$‎X‎$‎ contains a finite subset ‎‎$\mathcal{D}‎‎‎$‎ such that ‎‎$‎\bigcup \{‎\overline{‎D‎}‎\mid D ‎\in \mathcal{‎D}‎\} ‎‎=‎X‎‎$‎, i.e., the closures of the sets from ‎‎$‎‎\mathcal{D}‎$‎‎ cover ‎‎$‎X‎$‎.‎ ‎ ‎

‎I'm not totally fluent in the ‎star -‎compact space, and I'm investigating the ‎‎$‎H‎$‎-closed space. With regard to theorems 1,2, is there a relationship between the two spaces mentioned above?

I am looking for the link between these two ‎spaces (‎ I‎ ‎mean ‎star-‎ ‎compact ‎and ‎‎$‎H‎$‎-closed ‎spaces) if there is an article in this ‎field, can anyone introduce me to ‎it?‎

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It is well-known that a countably compact space is star-finite so certainly star-compact (see Brian's answer here for references, so there are many star-compact spaces that are not $H$-closed, e.g. $\omega_1$ in the order topology is such an example. If $X$ is $H$-closed it's going to be close to compact, but it's clearly strongly $2$-starcompact (in the DDRT terms; see the paper in Brian's answer) (the single star is dense). Not yet sure of we can conclude $H$-compact implies star-compact (in your sense). Of course yes for regular spaces..

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  • $\begingroup$ can you help me about this statement? " If X is H-closed it's going to be close to compact, but it's clearly strongly 2-starcompact". $\endgroup$ – adin Jun 19 at 12:01
  • $\begingroup$ Do you mean a $H$-closed is strongly 2-starcompact ? I can not find how $H$-closed space implies strongly 2-starcompact in DDRT terms.please help me.Thank you. $\endgroup$ – adin Jun 19 at 12:07
  • $\begingroup$ @adin that's straight from the definitions and the alternative characterisation of $H$-closedness. $\endgroup$ – Henno Brandsma Jun 19 at 15:25

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