‎‎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 $$X = \mathrm{st}(K,\mathscr{U}),$$ where $$\mathrm{st}(K, \mathscr{U}) = \bigcup \{ U \in \mathscr{U}: U \cap K \neq \emptyset \}.$$

We recursively define $$\mathrm{st}^n$$ for $$n=0,1,2,\ldots$$ by \begin{align} \mathrm{st}^0(K, \mathscr{U}) &= K \\ \mathrm{st}^{n+1}(K, \mathscr{U}‎) &= \bigcup ‎\{ U‎ ‎\in‎ \mathscr{‎U} : U ‎\cap\ \mathrm{st}^n(K, \mathscr{U}‎) \neq ‎\emptyset \}‎‎‎ \end{align}

Definition: ‎A space $$X$$ is said to be ‎$$‎\omega‎$$‎-starcompact if for every open cover ‎$$\mathscr{U}‎$$‎ of ‎‎$$X$$‎, there is some ‎‎$$‎n ‎\in ‎\mathbb{N‎}‎‎^{+}‎$$‎ and some finite subset ‎‎$$B$$‎ of ‎‎$$X$$‎ such that ‎‎$$\mathrm{st}^{‎n‎}(B, \mathscr{U}) = X‎$$‎.‎‎ ‎

Let‎‎ ‎$$(X, ‎\tau)$$ be‎ $$\omega‎$$‎-‎starcompact and $$\tau^{*} \subset \tau$$. Is $$(X, \tau^{*})$$ a‎ $$‎\omega‎$$‎-‎starcompact ‎space?‎

• Are $St$ and $st$ supposed to be the same? What is the base case in the recursive definition of $st^{n+1}$? Is $st^1 = St$? – md2perpe Jul 10 at 19:16
• I made an edit, primarily to fix some rendering errors. I found that there were a lot of Unicode left-to-right marks in the text. Removing them helped. Secondarily, in the edit I also fixed (I hope) the definition of $\mathrm{st}^n$. – md2perpe Jul 10 at 20:12

To see this, fix $$\mathscr{U} \subseteq \tau^* \subseteq \tau$$, an open cover of $$X$$, and pick a finite set $$B \subseteq X$$ and whole number $$n$$ for which $$st^n(B,\mathscr{U}) = X$$.
Note $$st(B,\mathscr{U})$$ is the same open set (i.e. the same union of open sets from the open cover) regardless of which topology we consider. In fact, for all $$k$$, $$st^k(B,\mathscr{U})$$ is the same union of open sets from the open cover regardless of which topology we consider. As such, $$st^n(B,\mathscr{U}) = X$$, regardless of which topology we consider.
Of course, every $$\tau^\ast$$-open cover is a $$\tau$$-open cover and we have $$B$$ there which still works. Many simple covering properties are preserved by going to a coarser topology, like (countable) compactness and Lindelöfness.