Theorems · Theorem · commutative algebra

Valuation.isClosed_sphere

∀ {R : Type u_1} [inst : Ring R] [inst_1 : ValuativeRel R] {Γ₀ : Type u_3} [inst_2 : LinearOrderedCommGroupWithZero Γ₀]
  [_t : TopologicalSpace R] [IsValuativeTopology R] {v : Valuation R Γ₀} [v.Compatible]
  (r : (MonoidWithZeroHom.ofClass v).ValueGroup₀), IsClosed {x | v.restrict x = r}

For any valuation v compatible with the valuative relation on R, the sphere of radius r around zero {x | v.restrict x = r} is closed in the valuative topology.

Defined in

Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Cited by

0 results in Mathlib

Foundations

Depth 105 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingValuativeRelLinearOrderedCommGroupWithZeroTopologicalSpaceIsValuativeTopologyValuation.Compatible