ValuativeTopology.lean`

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4 declarations, 0 contain sorry, 0 declared as axioms

Valuative topologies

Material destined for Mathlib.

Imports (2)

Imported by 2 files

instance (instance at line 30) line 30 source
Source documentation
A ring with a valuative topology is nonarchimedean. Mathlib proves this for the valuation-induced topology (ValuativeRel.nonarchimedeanRing, a local instance there) and registers the weaker transported instance IsValuativeTopology.isTopologicalRing, but not this nonarchimedean strengthening; transport it along Valuation.toTopologicalSpace_eq in the same way.
```
instance (priority := low) IsValuativeTopology.nonarchimedeanRing : NonarchimedeanRing R := by
…
```
instance (instance at line 42) line 42 source
Source documentation
A division ring with a valuative topology is Hausdorff: {y : |y| < |x|} is a neighbourhood of 0 not containing x. (Mathlib has this for Valued fields, ValuedRing.separated, but not for IsValuativeTopology.)
```
instance : T2Space K := by
…
```
def ValuativeRel.integerBallIdeal Documented declaration: integerBallIdeal line 53 source
Source documentation
The open ball {x ∈ 𝒪[K] | v(x) < γ} of the ring of integers is an ideal: ultrametrically it is closed under addition, and multiplying by an integral element only shrinks the valuation.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DivisionRing, ValuativeRel, Units, ValuativeRel.ValueGroupWithZero, Ideal, Subtype, Membership.mem, Subring, Valuation.integer, ValuativeRel.valuation
Proof uses: Submodule.mk, AddSubmonoid.mk, AddSubsemigroup.mk, Set.ofPred, LT.lt, Valuation, Set
44 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
def ValuativeRel.integerBallIdeal (γ : (ValueGroupWithZero K)ˣ) :
    Ideal (valuation K).integer where
…
```
instance (instance at line 73) line 73 source
Source documentation
The ring of integers of a valuative topology is linearly topologized: the open balls integerBallIdeal are ideals and form a basis of neighbourhoods of 0. This is the hypothesis under which mathlib's topological power-series evaluation (PowerSeries.eval₂Hom) applies to 𝒪[K] — it can never apply to the field K itself, which has no nontrivial ideals.
```
instance : IsLinearTopology (valuation K).integer (valuation K).integer := by
…
```