A non-2-starcompact, pseudocompact Tychonoff space whose hyperspace is 2-starcompact (Q1928421)

In this paper the star covering properties for hyperspaces (of nonempty closed subsets and of nonempty compact subsets) of a topological space \(X\) are investigated. The space \(X\) is assumed at least \(T_1\) and the hyperspace is equipped with the Vietoris topology. The author studies for which properties \(P\) and for which spaces \(X\) the implication (*) holds: (*) \textit{If the hyperspace of the space \(X\) has the property \(P\), then the space \(X\) has the property \(P\) too.} In particular it is proved that the implication (*): - is true for \(P = 1\)\textit{-starcompact } and for the hyperspace of closed subsets of a regular space; - is true for \(P = 1\frac{1}{2}\)\textit{-starcompact } and for the hyperspace of compact subsets; - is not true for \(P = 2\)\textit{-starcompact } and for the hyperspace of closed subsets of a pseudocompact Tychonoff space \(X\). The space \(X\) is the same which was constructed by \textit{I. J. Tree } in [Topology Appl. 47, No. 2, 129--132 (1992; Zbl 0766.54020)].