On topologies on the space of valuations and the valuative tree

TL;DR

This paper explores the topological structure of the space of valuations and the valuative tree, analyzing their properties and relationships using various topologies and subset characterizations.

Contribution

It establishes connections between different topologies on the valuative tree and characterizes its closure properties within a product space.

Findings

The weak tree topology and Scott topology are related in the valuative tree.

Supremums of increasing valuation families are described in specific subtrees.

The valuative tree is shown to be closed in the product topology of $(\Lambda_ abla)^{K[x]}$.

Abstract

In this paper, we discuss topological aspects of the space of valuations $\mathbb{V}$ and the valuative tree $\mathcal{T}(v,\Lambda)$. We present a relation between the weak tree topology and the Scott topology in $\mathcal{T}(v,\Lambda)$ and describe the supremum of an increasing family of valuations in a special subtree. We also view the valuative tree as a subset of the product $(\Lambda_\infty)^{K[x]}$ and prove that it is closed if we consider the natural product topology.