Cellular-compact and cellular-Lindelöf on hyperspaces

L.Q. Tuyen; O.V. Tuyen; N.X. Truc

doi:10.15421/242505

Cellular-compact and cellular-Lindelöf on hyperspaces

L.Q. Tuyen (Da Nang University of Education)
O.V. Tuyen (Hoa Vang high school), https://orcid.org/0000-0002-0232-2092
N.X. Truc (Da Nang University of Education)

Abstract

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 $$$\texttt{PR}[X]$$$ is cellular-compact (resp., cellular-Lindelöf) if and only if $$$X$$$ is finite (resp., countable). Moreover, if $$$\mathbb K(X)$$$ or $$$\mathcal F(X)$$$ or $$$\mathcal F_n(X)$$$ ($$$ n\in\mathbb N$$$) is cellular-compact (resp., cellular-Lindelöf), then $$$X$$$ is cellular-compact (resp., cellular-Lindelöf).

Keywords

hyperspace; Pixley-Roy topology; Vietoris topology; cellular-compact; cellular-Lindelöf

MSC 2020

54B20; 54D20; 54D30

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