Vietoris hyperspaces of scattered Priestley spaces

TL;DR

This paper investigates the properties of Vietoris hyperspaces of Priestley spaces, focusing on Skula topologies, their ordinal ranks, and applications to scattered compact spaces derived from special almost disjoint families.

Contribution

It provides new insights into when Vietoris hyperspaces of Priestley spaces are Skula and computes their ordinal ranks, extending understanding of their topological and order-theoretic structure.

Findings

Vietoris hyperspaces of Priestley spaces can be Skula under certain conditions.

Ordinal ranks of these hyperspaces are explicitly computed.

Applications include analysis of scattered compact spaces from Lusin families and ladder systems.

Abstract

We study Vietoris hyperspaces of closed and closed final sets of Priestley spaces. We are particularly interested in Skula topologies. A topological space is \emph{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.