Theorems · Theorem · commutative algebra

Valuation.isClopen_ball

∀ {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), IsClopen {x | v.restrict x < r}

For any valuation v compatible with the valuative relation on R, the open r-ball around zero {x | v.restrict x < r} is clopen in the valuative topology.

Defined in

Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Cited by

2 results in Mathlib

Foundations

Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingValuativeRelLinearOrderedCommGroupWithZeroTopologicalSpaceIsValuativeTopologyValuation.Compatible