Order convergence and compactness

TL;DR

This paper investigates the relationship between compact topologies finer than the interval topology and the order topology on partially ordered sets, showing that such topologies are contained within the order topology, including Priestley topologies.

Contribution

It establishes that any compact topology finer than the interval topology on a poset is contained in the order topology, extending understanding of order convergence and compactness.

Findings

Any compact topology finer than the interval topology is contained in the order topology.

Priestley topologies are contained in the order topology.

Provides conditions linking compactness, order convergence, and topology inclusion.

Abstract

Let $(P,\leq)$ be a partially ordered set and let $\tau$ be a compact topology on $P$ that is finer than the interval topology. Then $\tau$ is contained in the order (convergence) topology on $(P,\tau)$. So any Priestley topology is contained in the order topology.