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Vietoris hyperspaces over scattered Priestley spaces

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Abstract

We study Vietoris hyperspaces of closed and final sets of Priestley spaces. We are particularly interested in Skula topologies. A topological space is Skula if its topology is generated by differences of open sets of another topology. A compact Skula space is scattered and moreover has a natural well-founded ordering compatible with the topology, namely, it is a Priestley space. One of our main objectives is investigating Vietoris hyperspaces of general Priestley spaces, addressing the question when their topologies are Skula and computing the associated ordinal ranks. We apply our results to scattered compact spaces based on certain almost disjoint families, in particular, Lusin families and ladder systems.

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Authors and Affiliations

  1. Department of Algebra, topology and fundamentals of mathematics, Faculty of Mechanics and Mathematics, Ivan Franko National University of Lviv, Universytetska Street, 1, Lviv, 79000, Ukraine

    Taras Banakh

  2. Jan Kochanowski University, ul. Żeromskiego 5, 25-369, Kielce, Poland

    Taras Banakh

  3. Laboratoire de Mathématiques (LAMA), Université de Savoie Mont Blanc, 73376, Le Bourget-du-Lac, France

    Robert Bonnet

  4. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67, Praha 1, Czech Republic

    Wiesław Kubiś

  5. Cardinal Stefan Wyszynski University in Warsaw, Dewajtis 5, 01-815, Warsaw, Poland

    Wiesław Kubiś

Authors
  1. Taras Banakh
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  2. Robert Bonnet
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  3. Wiesław Kubiś
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Correspondence to Robert Bonnet.

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Dedicated to the memory of Matatyahu Rubin (1946–2017)

Partially supported by NCN grant DEC-2012/07/D/ST1/02087 (The National Science Center, Poland).

Supported by the Institute of Mathematics, Czech Academy of Sciences.

Supported by the GA ČR grant 20-22230L (Czech Science Foundation).

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Banakh, T., Bonnet, R. & Kubiś, W. Vietoris hyperspaces over scattered Priestley spaces. Isr. J. Math. 249, 37–81 (2022). https://doi.org/10.1007/s11856-022-2307-5

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  • DOI: https://doi.org/10.1007/s11856-022-2307-5

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