Two topologies on the lattice of Scott closed subsets

TL;DR

This paper explores the relationships between different topologies on the lattice of Scott closed subsets of a poset, providing conditions for their equivalence and addressing questions about core-compactness, sobriety, and consonance in power spaces.

Contribution

It introduces conditions under which the Scott and lower Vietoris topologies on the lattice of Scott closed subsets are equal and establishes an adjunction between related topologies, answering a prior open question.

Findings

Identified conditions for the equality of Scott and lower Vietoris topologies.

Proved that core-compactness, sobriety, and local compactness are equivalent under certain conditions.

Provided a partial answer to the question about consonance preservation in lower powerspaces.

Abstract

For a poset $P$, let $\sigma(P)$ and $\Gamma(P)$ respectively denote the lattice of its Scott open subsets and Scott closed subsets ordered by inclusion, and set $\Sigma P=(P,\sigma(P))$. In this paper, we discuss the lower Vietoris topology and the Scott topology on $\Gamma(P)$ and give some sufficient conditions to make the two topologies equal. We built an adjunction between $\sigma(P)$ and $\sigma(\Gamma(P))$ and proved that $\Sigma P$ is core-compact iff $\Sigma\Gamma(P)$ is core-compact iff $\Sigma\Gamma(P)$ is sober, locally compact and $\sigma(\Gamma(P))=\upsilon(\Gamma(P))$ (the lower Vietoris topology). This answers a question in [17]. Brecht and Kawai [2] asked whether the consonance of a topological space $X$ implies the consonance of its lower powerspace, we give a partial answer to this question at the last part of this paper.