What is the $\omega$- starcompact Complement?

A topological property ‎‎$$‎P‎$$‎ is expansive(contractive) if for any topology ‎$$‎\tau‎$$‎ with property ‎‎$$‎P‎$$‎ any finer (coarser) topology ‎$$‎\tau‎^{*}‎‎$$‎ also has property ‎‎‎$$P$$‎‎.‎ If ‎‎$$‎P$$‎ is an expansive topological property and ‎‎$$‎Q‎$$‎ is a contractive topological property then ‎‎$$‎P$$‎ and ‎‎$$‎Q‎$$‎ are called complementary if a topology is minimal ‎‎$$‎P$$‎ if and only if it is maximal ‎‎$$‎Q‎$$‎.‎‎‎

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

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

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

‎‎‎$$‎st‎^{n+1}‎‎ (K, \mathscr{U}‎) =‎ ‎\bigcup ‎\{ U‎ ‎\in‎ \mathscr{‎U} : U ‎\cap st‎^{n}‎‎‎(K, \mathscr{U}‎) \} ‎\neq ‎\emptyset ‎‎‎$$‎ ‎‎‎‎‎

I have little information about $$‎\omega‎$$‎-‎starcompact space and I'm having trouble connecting this space with other spaces.

‎1:‎Can anyone help me and ‎say, ‎What is the ‎‎$$‎\omega‎$$‎-‎starcompact ‎complement?‎‎

‎2: ‎I‎s ‎closed ‎subset ‎of‎ $$‎\omega‎$$‎-‎starcompact,‎ $$‎\omega‎$$‎-‎starcompact?‎

• As to 2.: how do you prove that a closed subspace of a compact is compact? Reapply that idea. – Henno Brandsma Jul 12 at 22:08
• You're mixing the older van Douwen style naming of these star-properties with the more recent one. If the first one is called star-compact, the last one should be something like $\omega$-star-finite. Less confusing. I don't think there is an implication either way between the two properties defined in the post... – Henno Brandsma Jul 12 at 22:11