# a question about star compact space

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?‎

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..
• 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. – adin Jun 19 at 12:07
• @adin that's straight from the definitions and the alternative characterisation of $H$-closedness. – Henno Brandsma Jun 19 at 15:25