This analysis addresses cellular-compact and cellular-Lindelöf properties in hyperspaces, indicating essential conditions for finite and countable spaces.
The generalized metric properties on hyperspaces with the Pixley-Roy topology and the Vietoris topology have been studied by many authors. They considered several generalized metric properties and studied the relation between a space X satisfying such property and its hyperspaces with the Pixley-Roy topology and the Vietoris topology satisfying the same property. In this paper, we study cellular-compact and cellular-Lindelöf spaces on hyperspaces with the Pixley-Roy topology and the Vietoris topology. For a space X, we prove that PR[X] is cellular-compact (resp., cellular-Lindelöf) if and only if X is finite (resp., countable). Moreover, if K(X) or F(X) or Fₙ(X) (n∈ N) is cellular-compact (resp., cellular-Lindelöf), then X is cellular-compact (resp., cellular-Lindelöf).
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Tuyen et al. (2025) studied this question.
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