Topological (semi)rings

A topological (semi)ring is a (semi)ring equipped with a topology such that all operations are continuous. Besides this definition, this file proves that the topological closure of a subring (resp. an ideal) is a subring (resp. an ideal) and defines products and quotients of topological (semi)rings.

Main Results

• Subring.topologicalClosure/Subsemiring.topologicalClosure: the topological closure of a Subring/Subsemiring is itself a Sub(semi)ring.
• The product of two topological (semi)rings is a topological (semi)ring.
• The indexed product of topological (semi)rings is a topological (semi)ring.

class IsTopologicalSemiring (R : Type u_1) [TopologicalSpace R] [NonUnitalNonAssocSemiring R] extends ContinuousAdd R, ContinuousMul R : Prop

a topological semiring is a semiring R where addition and multiplication are continuous. We allow for non-unital and non-associative semirings as well.

The IsTopologicalSemiring class should only be instantiated in the presence of a NonUnitalNonAssocSemiring instance; if there is an instance of NonUnitalNonAssocRing, then IsTopologicalRing should be used. Note: in the presence of NonAssocRing, these classes are mathematically equivalent (see IsTopologicalSemiring.continuousNeg_of_mul or IsTopologicalSemiring.toIsTopologicalRing).

• continuous_add : Continuous fun (p : R × R) => p.1 + p.2
• continuous_mul : Continuous fun (p : R × R) => p.1 * p.2

class IsTopologicalRing (R : Type u_1) [TopologicalSpace R] [NonUnitalNonAssocRing R] extends IsTopologicalSemiring R, ContinuousNeg R : Prop

A topological ring is a ring R where addition, multiplication and negation are continuous.

If R is a (unital) ring, then continuity of negation can be derived from continuity of multiplication as it is multiplication with -1. (See IsTopologicalSemiring.continuousNeg_of_mul and topological_semiring.to_topological_add_group)

• continuous_add : Continuous fun (p : R × R) => p.1 + p.2
• continuous_mul : Continuous fun (p : R × R) => p.1 * p.2
• continuous_neg : Continuous fun (a : R) => -a

theorem IsTopologicalSemiring.continuousNeg_of_mul {R : Type u_1} [TopologicalSpace R] [NonAssocRing R] [ContinuousMul R] : ContinuousNeg R

If R is a ring with a continuous multiplication, then negation is continuous as well since it is just multiplication with -1.

theorem IsTopologicalSemiring.toIsTopologicalRing {R : Type u_1} [TopologicalSpace R] [NonAssocRing R] : IsTopologicalSemiring R → IsTopologicalRing R

If R is a ring which is a topological semiring, then it is automatically a topological ring. This exists so that one can place a topological ring structure on R without explicitly proving continuous_neg.

@[instance 100]

instance IsTopologicalRing.to_topologicalAddGroup {R : Type u_1} [NonUnitalNonAssocRing R] [TopologicalSpace R] [IsTopologicalRing R] : IsTopologicalAddGroup R

@[instance 50]

instance DiscreteTopology.topologicalSemiring {R : Type u_1} [TopologicalSpace R] [NonUnitalNonAssocSemiring R] [DiscreteTopology R] : IsTopologicalSemiring R

@[instance 50]

instance DiscreteTopology.topologicalRing {R : Type u_1} [TopologicalSpace R] [NonUnitalNonAssocRing R] [DiscreteTopology R] : IsTopologicalRing R

instance NonUnitalSubsemiring.instIsTopologicalSemiring {R : Type u_1} [TopologicalSpace R] [NonUnitalSemiring R] [IsTopologicalSemiring R] (S : NonUnitalSubsemiring R) : IsTopologicalSemiring ↥S

def NonUnitalSubsemiring.topologicalClosure {R : Type u_1} [TopologicalSpace R] [NonUnitalSemiring R] [IsTopologicalSemiring R] (s : NonUnitalSubsemiring R) : NonUnitalSubsemiring R

The (topological) closure of a non-unital subsemiring of a non-unital topological semiring is itself a non-unital subsemiring.

Equations

• s.topologicalClosure = { carrier := closure ↑s, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ }

@[simp]

theorem NonUnitalSubsemiring.topologicalClosure_coe {R : Type u_1} [TopologicalSpace R] [NonUnitalSemiring R] [IsTopologicalSemiring R] (s : NonUnitalSubsemiring R) : ↑s.topologicalClosure = _root_.closure ↑s

theorem NonUnitalSubsemiring.le_topologicalClosure {R : Type u_1} [TopologicalSpace R] [NonUnitalSemiring R] [IsTopologicalSemiring R] (s : NonUnitalSubsemiring R) : s ≤ s.topologicalClosure

theorem NonUnitalSubsemiring.isClosed_topologicalClosure {R : Type u_1} [TopologicalSpace R] [NonUnitalSemiring R] [IsTopologicalSemiring R] (s : NonUnitalSubsemiring R) : IsClosed ↑s.topologicalClosure

theorem NonUnitalSubsemiring.topologicalClosure_minimal {R : Type u_1} [TopologicalSpace R] [NonUnitalSemiring R] [IsTopologicalSemiring R] (s : NonUnitalSubsemiring R) {t : NonUnitalSubsemiring R} (h : s ≤ t) (ht : IsClosed ↑t) : s.topologicalClosure ≤ t

@[reducible, inline]

abbrev NonUnitalSubsemiring.nonUnitalCommSemiringTopologicalClosure {R : Type u_1} [TopologicalSpace R] [NonUnitalSemiring R] [IsTopologicalSemiring R] [T2Space R] (s : NonUnitalSubsemiring R) (hs : ∀ (x y : ↥s), x * y = y * x) : NonUnitalCommSemiring ↥s.topologicalClosure

If a non-unital subsemiring of a non-unital topological semiring is commutative, then so is its topological closure.

See note [reducible non-instances]

Equations

• s.nonUnitalCommSemiringTopologicalClosure hs = { toNonUnitalSemiring := NonUnitalSubsemiringClass.toNonUnitalSemiring s.topologicalClosure, mul_comm := ⋯ }

instance instIsTopologicalSemiringULift {R : Type u_1} [TopologicalSpace R] [Semiring R] [IsTopologicalSemiring R] : IsTopologicalSemiring (ULift.{u_2, u_1} R)

instance Subsemiring.topologicalSemiring {R : Type u_1} [TopologicalSpace R] [Semiring R] [IsTopologicalSemiring R] (S : Subsemiring R) : IsTopologicalSemiring ↥S

instance Subsemiring.continuousSMul {R : Type u_1} [TopologicalSpace R] [Semiring R] (s : Subsemiring R) (X : Type u_2) [TopologicalSpace X] [MulAction R X] [ContinuousSMul R X] : ContinuousSMul (↥s) X

def Subsemiring.topologicalClosure {R : Type u_1} [TopologicalSpace R] [Semiring R] [IsTopologicalSemiring R] (s : Subsemiring R) : Subsemiring R

The (topological-space) closure of a subsemiring of a topological semiring is itself a subsemiring.

Equations

• s.topologicalClosure = { carrier := closure ↑s, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯ }

@[simp]

theorem Subsemiring.topologicalClosure_coe {R : Type u_1} [TopologicalSpace R] [Semiring R] [IsTopologicalSemiring R] (s : Subsemiring R) : ↑s.topologicalClosure = _root_.closure ↑s

theorem Subsemiring.le_topologicalClosure {R : Type u_1} [TopologicalSpace R] [Semiring R] [IsTopologicalSemiring R] (s : Subsemiring R) : s ≤ s.topologicalClosure

theorem Subsemiring.isClosed_topologicalClosure {R : Type u_1} [TopologicalSpace R] [Semiring R] [IsTopologicalSemiring R] (s : Subsemiring R) : IsClosed ↑s.topologicalClosure

theorem Subsemiring.topologicalClosure_minimal {R : Type u_1} [TopologicalSpace R] [Semiring R] [IsTopologicalSemiring R] (s : Subsemiring R) {t : Subsemiring R} (h : s ≤ t) (ht : IsClosed ↑t) : s.topologicalClosure ≤ t

@[reducible, inline]

abbrev Subsemiring.commSemiringTopologicalClosure {R : Type u_1} [TopologicalSpace R] [Semiring R] [IsTopologicalSemiring R] [T2Space R] (s : Subsemiring R) (hs : ∀ (x y : ↥s), x * y = y * x) : CommSemiring ↥s.topologicalClosure

If a subsemiring of a topological semiring is commutative, then so is its topological closure.

See note [reducible non-instances].

Equations

• s.commSemiringTopologicalClosure hs = { toSemiring := s.topologicalClosure.toSemiring, mul_comm := ⋯ }

instance instIsTopologicalSemiringProd {R : Type u_1} {S : Type u_2} [TopologicalSpace R] [TopologicalSpace S] [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [IsTopologicalSemiring R] [IsTopologicalSemiring S] : IsTopologicalSemiring (R × S)

The product topology on the Cartesian product of two topological semirings makes the product into a topological semiring.

instance instIsTopologicalRingProd {R : Type u_1} {S : Type u_2} [TopologicalSpace R] [TopologicalSpace S] [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [IsTopologicalRing R] [IsTopologicalRing S] : IsTopologicalRing (R × S)

The product topology on the Cartesian product of two topological rings makes the product into a topological ring.

instance instContinuousAddForallOfIsTopologicalSemiring {ι : Type u_2} {R : ι → Type u_3} [(i : ι) → TopologicalSpace (R i)] [(i : ι) → NonUnitalNonAssocSemiring (R i)] [∀ (i : ι), IsTopologicalSemiring (R i)] : ContinuousAdd ((i : ι) → R i)

instance Pi.instIsTopologicalSemiring {ι : Type u_2} {R : ι → Type u_3} [(i : ι) → TopologicalSpace (R i)] [(i : ι) → NonUnitalNonAssocSemiring (R i)] [∀ (i : ι), IsTopologicalSemiring (R i)] : IsTopologicalSemiring ((i : ι) → R i)

instance Pi.instIsTopologicalRing {ι : Type u_2} {R : ι → Type u_3} [(i : ι) → TopologicalSpace (R i)] [(i : ι) → NonUnitalNonAssocRing (R i)] [∀ (i : ι), IsTopologicalRing (R i)] : IsTopologicalRing ((i : ι) → R i)

instance instContinuousAddMulOpposite {R : Type u_1} [NonUnitalNonAssocSemiring R] [TopologicalSpace R] [ContinuousAdd R] : ContinuousAdd Rᵐᵒᵖ

instance instIsTopologicalSemiringMulOpposite {R : Type u_1} [NonUnitalNonAssocSemiring R] [TopologicalSpace R] [IsTopologicalSemiring R] : IsTopologicalSemiring Rᵐᵒᵖ

instance instContinuousNegMulOpposite {R : Type u_1} [NonUnitalNonAssocRing R] [TopologicalSpace R] [ContinuousNeg R] : ContinuousNeg Rᵐᵒᵖ

instance instIsTopologicalRingMulOpposite {R : Type u_1} [NonUnitalNonAssocRing R] [TopologicalSpace R] [IsTopologicalRing R] : IsTopologicalRing Rᵐᵒᵖ

instance instContinuousMulAddOpposite {R : Type u_1} [NonUnitalNonAssocSemiring R] [TopologicalSpace R] [ContinuousMul R] : ContinuousMul Rᵃᵒᵖ

instance instIsTopologicalSemiringAddOpposite {R : Type u_1} [NonUnitalNonAssocSemiring R] [TopologicalSpace R] [IsTopologicalSemiring R] : IsTopologicalSemiring Rᵃᵒᵖ

instance instIsTopologicalRingAddOpposite {R : Type u_1} [NonUnitalNonAssocRing R] [TopologicalSpace R] [IsTopologicalRing R] : IsTopologicalRing Rᵃᵒᵖ

theorem IsTopologicalRing.of_addGroup_of_nhds_zero {R : Type u_2} [NonUnitalNonAssocRing R] [TopologicalSpace R] [IsTopologicalAddGroup R] (hmul : Filter.Tendsto (Function.uncurry fun (x1 x2 : R) => x1 * x2) (nhds 0 ×ˢ nhds 0) (nhds 0)) (hmul_left : ∀ (x₀ : R), Filter.Tendsto (fun (x : R) => x₀ * x) (nhds 0) (nhds 0)) (hmul_right : ∀ (x₀ : R), Filter.Tendsto (fun (x : R) => x * x₀) (nhds 0) (nhds 0)) : IsTopologicalRing R

theorem IsTopologicalRing.of_nhds_zero {R : Type u_2} [NonUnitalNonAssocRing R] [TopologicalSpace R] (hadd : Filter.Tendsto (Function.uncurry fun (x1 x2 : R) => x1 + x2) (nhds 0 ×ˢ nhds 0) (nhds 0)) (hneg : Filter.Tendsto (fun (x : R) => -x) (nhds 0) (nhds 0)) (hmul : Filter.Tendsto (Function.uncurry fun (x1 x2 : R) => x1 * x2) (nhds 0 ×ˢ nhds 0) (nhds 0)) (hmul_left : ∀ (x₀ : R), Filter.Tendsto (fun (x : R) => x₀ * x) (nhds 0) (nhds 0)) (hmul_right : ∀ (x₀ : R), Filter.Tendsto (fun (x : R) => x * x₀) (nhds 0) (nhds 0)) (hleft : ∀ (x₀ : R), nhds x₀ = Filter.map (fun (x : R) => x₀ + x) (nhds 0)) : IsTopologicalRing R

instance instIsTopologicalRingULift {R : Type u_1} [TopologicalSpace R] [NonUnitalNonAssocRing R] [IsTopologicalRing R] : IsTopologicalRing (ULift.{u_2, u_1} R)

theorem mulLeft_continuous {R : Type u_1} [TopologicalSpace R] [NonUnitalNonAssocRing R] [IsTopologicalRing R] (x : R) : Continuous ⇑(AddMonoidHom.mulLeft x)

In a topological semiring, the left-multiplication AddMonoidHom is continuous.

theorem mulRight_continuous {R : Type u_1} [TopologicalSpace R] [NonUnitalNonAssocRing R] [IsTopologicalRing R] (x : R) : Continuous ⇑(AddMonoidHom.mulRight x)

In a topological semiring, the right-multiplication AddMonoidHom is continuous.

instance NonUnitalSubring.instIsTopologicalRing {R : Type u_1} [TopologicalSpace R] [NonUnitalRing R] [IsTopologicalRing R] (S : NonUnitalSubring R) : IsTopologicalRing ↥S

def NonUnitalSubring.topologicalClosure {R : Type u_1} [TopologicalSpace R] [NonUnitalRing R] [IsTopologicalRing R] (S : NonUnitalSubring R) : NonUnitalSubring R

The (topological) closure of a non-unital subring of a non-unital topological ring is itself a non-unital subring.

Equations

• S.topologicalClosure = { carrier := closure ↑S, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯, neg_mem' := ⋯ }

theorem NonUnitalSubring.le_topologicalClosure {R : Type u_1} [TopologicalSpace R] [NonUnitalRing R] [IsTopologicalRing R] (s : NonUnitalSubring R) : s ≤ s.topologicalClosure

theorem NonUnitalSubring.isClosed_topologicalClosure {R : Type u_1} [TopologicalSpace R] [NonUnitalRing R] [IsTopologicalRing R] (s : NonUnitalSubring R) : IsClosed ↑s.topologicalClosure

theorem NonUnitalSubring.topologicalClosure_minimal {R : Type u_1} [TopologicalSpace R] [NonUnitalRing R] [IsTopologicalRing R] (s : NonUnitalSubring R) {t : NonUnitalSubring R} (h : s ≤ t) (ht : IsClosed ↑t) : s.topologicalClosure ≤ t

@[reducible, inline]

abbrev NonUnitalSubring.nonUnitalCommRingTopologicalClosure {R : Type u_1} [TopologicalSpace R] [NonUnitalRing R] [IsTopologicalRing R] [T2Space R] (s : NonUnitalSubring R) (hs : ∀ (x y : ↥s), x * y = y * x) : NonUnitalCommRing ↥s.topologicalClosure

If a non-unital subring of a non-unital topological ring is commutative, then so is its topological closure.

See note [reducible non-instances]

Equations

• s.nonUnitalCommRingTopologicalClosure hs = { toNonUnitalRing := s.topologicalClosure.toNonUnitalRing, mul_comm := ⋯ }

instance Subring.instIsTopologicalRing {R : Type u_1} [TopologicalSpace R] [Ring R] [IsTopologicalRing R] (S : Subring R) : IsTopologicalRing ↥S

instance Subring.continuousSMul {R : Type u_1} [TopologicalSpace R] [Ring R] (s : Subring R) (X : Type u_2) [TopologicalSpace X] [MulAction R X] [ContinuousSMul R X] : ContinuousSMul (↥s) X

def Subring.topologicalClosure {R : Type u_1} [TopologicalSpace R] [Ring R] [IsTopologicalRing R] (S : Subring R) : Subring R

The (topological-space) closure of a subring of a topological ring is itself a subring.

Equations

• S.topologicalClosure = { carrier := closure ↑S, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯, neg_mem' := ⋯ }

theorem Subring.le_topologicalClosure {R : Type u_1} [TopologicalSpace R] [Ring R] [IsTopologicalRing R] (s : Subring R) : s ≤ s.topologicalClosure

theorem Subring.isClosed_topologicalClosure {R : Type u_1} [TopologicalSpace R] [Ring R] [IsTopologicalRing R] (s : Subring R) : IsClosed ↑s.topologicalClosure

theorem Subring.topologicalClosure_minimal {R : Type u_1} [TopologicalSpace R] [Ring R] [IsTopologicalRing R] (s : Subring R) {t : Subring R} (h : s ≤ t) (ht : IsClosed ↑t) : s.topologicalClosure ≤ t

@[reducible, inline]

abbrev Subring.commRingTopologicalClosure {R : Type u_1} [TopologicalSpace R] [Ring R] [IsTopologicalRing R] [T2Space R] (s : Subring R) (hs : ∀ (x y : ↥s), x * y = y * x) : CommRing ↥s.topologicalClosure

If a subring of a topological ring is commutative, then so is its topological closure.

See note [reducible non-instances].

Equations

• s.commRingTopologicalClosure hs = { toRing := s.topologicalClosure.toRing, mul_comm := ⋯ }

Lattice of ring topologies

We define a type class RingTopology R which endows a ring R with a topology such that all ring operations are continuous.

Ring topologies on a fixed ring R are ordered, by reverse inclusion. They form a complete lattice, with ⊥ the discrete topology and ⊤ the indiscrete topology.

Any function f : R → S induces coinduced f : TopologicalSpace R → RingTopology S.

structure RingTopology (R : Type u) [Ring R] extends TopologicalSpace R, IsTopologicalRing R : Type u

A ring topology on a ring R is a topology for which addition, negation and multiplication are continuous.

• IsOpen : Set R → Prop
• isOpen_univ : TopologicalSpace.IsOpen Set.univ
• isOpen_inter (s t : Set R) : TopologicalSpace.IsOpen s → TopologicalSpace.IsOpen t → TopologicalSpace.IsOpen (s ∩ t)
• isOpen_sUnion (s : Set (Set R)) : (∀ t ∈ s, TopologicalSpace.IsOpen t) → TopologicalSpace.IsOpen (⋃₀ s)
• continuous_add : Continuous fun (p : R × R) => p.1 + p.2
• continuous_mul : Continuous fun (p : R × R) => p.1 * p.2
• continuous_neg : Continuous fun (a : R) => -a

instance RingTopology.inhabited {R : Type u} [Ring R] : Inhabited (RingTopology R)

Equations

• RingTopology.inhabited = { default := let x := ⊤; { toTopologicalSpace := x, toIsTopologicalRing := ⋯ } }

theorem RingTopology.toTopologicalSpace_injective {R : Type u_1} [Ring R] : Function.Injective toTopologicalSpace

theorem RingTopology.ext {R : Type u_1} [Ring R] {f g : RingTopology R} (h : TopologicalSpace.IsOpen = TopologicalSpace.IsOpen) : f = g

theorem RingTopology.ext_iff {R : Type u_1} [Ring R] {f g : RingTopology R} : f = g ↔ TopologicalSpace.IsOpen = TopologicalSpace.IsOpen

instance RingTopology.instPartialOrder {R : Type u_1} [Ring R] : PartialOrder (RingTopology R)

The ordering on ring topologies on the ring R. t ≤ s if every set open in s is also open in t (t is finer than s).

Equations

• RingTopology.instPartialOrder = PartialOrder.lift RingTopology.toTopologicalSpace ⋯

instance RingTopology.instCompleteSemilatticeInf {R : Type u_1} [Ring R] : CompleteSemilatticeInf (RingTopology R)

Ring topologies on R form a complete lattice, with ⊥ the discrete topology and ⊤ the indiscrete topology.

The infimum of a collection of ring topologies is the topology generated by all their open sets (which is a ring topology).

The supremum of two ring topologies s and t is the infimum of the family of all ring topologies contained in the intersection of s and t.

Equations

• RingTopology.instCompleteSemilatticeInf = { toPartialOrder := RingTopology.instPartialOrder, sInf := RingTopology.def_sInf✝, sInf_le := ⋯, le_sInf := ⋯ }

instance RingTopology.instCompleteLattice {R : Type u_1} [Ring R] : CompleteLattice (RingTopology R)

Equations

• RingTopology.instCompleteLattice = completeLatticeOfCompleteSemilatticeInf (RingTopology R)

def RingTopology.coinduced {R : Type u_2} {S : Type u_3} [t : TopologicalSpace R] [Ring S] (f : R → S) : RingTopology S

Given f : R → S and a topology on R, the coinduced ring topology on S is the finest topology such that f is continuous and S is a topological ring.

Equations

• RingTopology.coinduced f = sInf {b : RingTopology S | TopologicalSpace.coinduced f t ≤ b.toTopologicalSpace}

theorem RingTopology.coinduced_continuous {R : Type u_2} {S : Type u_3} [t : TopologicalSpace R] [Ring S] (f : R → S) : Continuous f

def RingTopology.toAddGroupTopology {R : Type u_1} [Ring R] (t : RingTopology R) : AddGroupTopology R

The forgetful functor from ring topologies on a to additive group topologies on a.

Equations

• t.toAddGroupTopology = { toTopologicalSpace := t.toTopologicalSpace, toIsTopologicalAddGroup := ⋯ }

def RingTopology.toAddGroupTopology.orderEmbedding {R : Type u_1} [Ring R] : RingTopology R ↪o AddGroupTopology R

The order embedding from ring topologies on a to additive group topologies on a.

Equations

• RingTopology.toAddGroupTopology.orderEmbedding = OrderEmbedding.ofMapLEIff RingTopology.toAddGroupTopology ⋯

def AbsoluteValue.comp {R : Type u_1} {S : Type u_2} {T : Type u_3} [Semiring T] [Semiring R] [Semiring S] [PartialOrder S] (v : AbsoluteValue R S) {f : T →+* R} (hf : Function.Injective ⇑f) : AbsoluteValue T S

Construct an absolute value on a semiring T from an absolute value on a semiring R and an injective ring homomorphism f : T →+* R

Equations

• v.comp hf = { toMulHom := v.comp ↑f, nonneg' := ⋯, eq_zero' := ⋯, add_le' := ⋯ }