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Relative versions of first and second countability in hyperspaces

  • Jesús Díaz-Reyes EMAIL logo and Jesús F. Tenorio
Published/Copyright: December 6, 2024
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 74 Issue 6

Abstract

Let CL(X) be the collection of all non-empty closed subsets of X, and Δ any subfamily of CL(X). By τΔ=τV−∨τΔ+ , we denote the hit-and-miss topology on CL(X). In 1995, Arhangel’skii focused his attention on the following general problem was formulated: given any subspace Y of X, study the subspace Y+ of CL(X) in the Vietoris topology and the other natural topologies, where Y+ is the subspace of CL(X) consisting of all non-empty closed subsets of X which are contained in Y. In this paper, we investigate when Y+ is first (second) countable in (CL(X), τ), for τ∈{τΔ,τV−,τΔ+} . All content of this work extend some results of the theory of hyperspaces, when Y=X or when Δ coincides with CL(X) or when is equal to the collection of all non-empty compact subsets of X.

MSC 2010: Primary 54B20, 54B05; Secondary 54D20, 54E45
Keywords: First countability; second countability; relative properties; hyperspaces; hit-and-miss topology

Funding statement: The first author was supported by “Programa de Estancias Posdoctorales 2021(1), CONAHCyT”.

Acknowledgement

The authors would like to thank the referee for careful reading and very useful comments and suggestions that contributed to improving the quality of this paper.

  1. Communicated by L’ubica Holá

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Received: 2024-02-13
Accepted: 2024-07-15
Published Online: 2024-12-06
Published in Print: 2024-12-15

© 2024 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2024-0113
Keywords for this article
First countability; second countability; relative properties; hyperspaces; hit-and-miss topology

Articles in the same Issue

  1. Algebraic structures formalizing the logic of effect algebras incorporating time dimension
  2. Walks on tiled boards
  3. Composition on FLew-algebras
  4. On high power sums of a hybrid arithmetic function
  5. On indices of quintic number fields defined by x5 + ax + b
  6. Note on fundamental system of solutions to the differential equations (D2 − 2Dα + α2 ± β2) y = 0
  7. Several sharp inequalities involving (hyperbolic) tangent, tanc, cosine, and their reciprocals
  8. A new Approach of Generalized Fractional Integrals in Multiplicative Calculus and Related Hermite–Hadamard-Type Inequalities with Applications
  9. On a periodic problem for super-linear second-order ODEs
  10. Existence and Uniqueness of Solutions for Fractional Dynamic Equations with Impulse Effects
  11. Periodic Solutions for Conformable Non-autonomous Non-instantaneous Impulsive Differential Equations
  12. Self referred equations with an integral boundary condition
  13. Approximation by matrix means of double Vilenkin-Fourier series
  14. Maps on self-adjoint operators preserving some relations related to commutativity
  15. Chains in the Rudin-Frolík order
  16. Relative versions of first and second countability in hyperspaces
  17. Proportion estimation in multistage pair ranked set sampling
  18. The Marshall-Olkin Extended Gamma Lindley distribution: Properties, characterization and inference
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