The Scott topology of a rooted non-metric tree

TL;DR

This paper investigates the relationship between the Scott topology and the weak tree topology on rooted non-metric trees, showing the Scott topology is strictly coarser and that the weak topology is generated by Scott topologies from various roots.

Contribution

It establishes the strict coarseness of the Scott topology compared to the weak tree topology and characterizes the weak topology as generated by Scott topologies from different root-based orders.

Findings

Scott topology is strictly coarser than the weak tree topology.

The weak tree topology is generated by union of Scott topologies from various roots.

Provides a new understanding of topological structures on non-metric trees.

Abstract

In this paper we prove that the Scott topology $\mathfrak S$ on a rooted non-metric tree $\mathcal T$ is strictly coarser than the weak tree topology. Moreover, for each $t\in \mathcal T$, we consider a natural order $\preceq_t$ on $\mathcal T$ under which $t$ is the root of $\mathcal T$. Then the weak tree topology is generated by the union of the Scott topologies $\mathfrak S_t$ associated to $\preceq_t$.