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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?‎

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  • $\begingroup$ As to 2.: how do you prove that a closed subspace of a compact is compact? Reapply that idea. $\endgroup$ – Henno Brandsma Jul 12 at 22:08
  • $\begingroup$ 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... $\endgroup$ – Henno Brandsma Jul 12 at 22:11

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