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..