The weak tree topology coincides with the Scott topology on the valuative tree, and the tree is closed in the natural product topology on (Λ_∞)^{K[x]}.
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On topologies on the space of valuations and the valuative tree
The weak tree topology coincides with the Scott topology on the valuative tree, and the tree is closed in the natural product topology on (Λ_∞)^{K[x]}.