On topologies on the space of valuations and the valuative tree

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.