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arXiv · 2604.02561

On topologies on the space of valuations and the valuative tree

Vinicius Manfredini·Josnei Novacoski·Caio Henrique Silva de Souza
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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.

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Vinicius Manfredini, Josnei Novacoski, Caio Henrique Silva de Souza. 2026-04-02. On topologies on the space of valuations and the valuative tree. https://arxiv.org/abs/2604.02561

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