Starcompact and related spaces on Pixley-Roy hyperspaces

. In this paper, we study the relation between a space X satisfying certain generalized metric properties and the Pixley-Roy hy-perspace PR [ X ] over X satisfying the same properties. We prove that if PR [ X ] is starcompact (resp., star-Lindel¨of), then X is compact (resp., Lindel¨of). However, there exists a compact space X such that | X | = ω , but PR n [ X ] for all n ∈ N and PR [ X ] are not starcompact spaces. More-over, we show that PR [ X ] is strongly starcompact (resp., strongly star-Lindel¨of) if and only if X is finite (resp., countable). By these results, we obtain that PR [ X ] is set strongly starcompact (resp., cosmic, set strongly star-Lindel¨of) if and only if X is finite (resp., countable).