Topology on TrivSqZeroExt R M

The type TrivSqZeroExt R M inherits the topology from R × M.

Note that this is not the topology induced by the seminorm on the dual numbers suggested by this Math.SE answer, which instead induces the topology pulled back through the projection map TrivSqZeroExt.fst : tsze R M → R. Obviously, that topology is not Hausdorff and using it would result in exp converging to more than one value.

Main results

• TrivSqZeroExt.topologicalRing: the ring operations are continuous

instance TrivSqZeroExt.instTopologicalSpace {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] : TopologicalSpace (TrivSqZeroExt R M)

Equations

• TrivSqZeroExt.instTopologicalSpace = TopologicalSpace.induced TrivSqZeroExt.fst inst✝¹ ⊓ TopologicalSpace.induced TrivSqZeroExt.snd inst✝

instance TrivSqZeroExt.instT2Space {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [T2Space R] [T2Space M] : T2Space (TrivSqZeroExt R M)

theorem TrivSqZeroExt.nhds_def {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] (x : TrivSqZeroExt R M) : nhds x = nhds x.fst ×ˢ nhds x.snd

theorem TrivSqZeroExt.nhds_inl {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Zero M] (x : R) : nhds (TrivSqZeroExt.inl x) = nhds x ×ˢ nhds 0

theorem TrivSqZeroExt.nhds_inr {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Zero R] (m : M) : nhds (TrivSqZeroExt.inr m) = nhds 0 ×ˢ nhds m

theorem TrivSqZeroExt.continuous_fst {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] : Continuous TrivSqZeroExt.fst

theorem TrivSqZeroExt.continuous_snd {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] : Continuous TrivSqZeroExt.snd

theorem TrivSqZeroExt.continuous_inl {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Zero M] : Continuous TrivSqZeroExt.inl

theorem TrivSqZeroExt.continuous_inr {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Zero R] : Continuous TrivSqZeroExt.inr

theorem TrivSqZeroExt.IsEmbedding.inl {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Zero M] : Topology.IsEmbedding TrivSqZeroExt.inl

@[deprecated TrivSqZeroExt.IsEmbedding.inl (since := "2024-10-26")]

theorem TrivSqZeroExt.embedding_inl {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Zero M] : Topology.IsEmbedding TrivSqZeroExt.inl

Alias of TrivSqZeroExt.IsEmbedding.inl.

theorem TrivSqZeroExt.IsEmbedding.inr {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Zero R] : Topology.IsEmbedding TrivSqZeroExt.inr

@[deprecated TrivSqZeroExt.IsEmbedding.inr (since := "2024-10-26")]

theorem TrivSqZeroExt.embedding_inr {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Zero R] : Topology.IsEmbedding TrivSqZeroExt.inr

Alias of TrivSqZeroExt.IsEmbedding.inr.

def TrivSqZeroExt.fstCLM (R : Type u_3) (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [CommSemiring R] [AddCommMonoid M] [Module R M] : TrivSqZeroExt R M →L[R] R

TrivSqZeroExt.fst as a continuous linear map.

Equations

• TrivSqZeroExt.fstCLM R M = { toFun := TrivSqZeroExt.fst, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ }

@[simp]

theorem TrivSqZeroExt.fstCLM_apply (R : Type u_3) (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [CommSemiring R] [AddCommMonoid M] [Module R M] (x : TrivSqZeroExt R M) : (TrivSqZeroExt.fstCLM R M) x = x.fst

def TrivSqZeroExt.sndCLM (R : Type u_3) (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [CommSemiring R] [AddCommMonoid M] [Module R M] : TrivSqZeroExt R M →L[R] M

TrivSqZeroExt.snd as a continuous linear map.

Equations

• TrivSqZeroExt.sndCLM R M = { toFun := TrivSqZeroExt.snd, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ }

@[simp]

theorem TrivSqZeroExt.sndCLM_apply (R : Type u_3) (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [CommSemiring R] [AddCommMonoid M] [Module R M] (x : TrivSqZeroExt R M) : (TrivSqZeroExt.sndCLM R M) x = x.snd

def TrivSqZeroExt.inlCLM (R : Type u_3) (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [CommSemiring R] [AddCommMonoid M] [Module R M] : R →L[R] TrivSqZeroExt R M

TrivSqZeroExt.inl as a continuous linear map.

Equations

• TrivSqZeroExt.inlCLM R M = { toFun := TrivSqZeroExt.inl, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ }

@[simp]

theorem TrivSqZeroExt.inlCLM_apply (R : Type u_3) (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [CommSemiring R] [AddCommMonoid M] [Module R M] (r : R) : (TrivSqZeroExt.inlCLM R M) r = TrivSqZeroExt.inl r

def TrivSqZeroExt.inrCLM (R : Type u_3) (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [CommSemiring R] [AddCommMonoid M] [Module R M] : M →L[R] TrivSqZeroExt R M

TrivSqZeroExt.inr as a continuous linear map.

Equations

• TrivSqZeroExt.inrCLM R M = { toFun := TrivSqZeroExt.inr, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ }

@[simp]

theorem TrivSqZeroExt.inrCLM_apply (R : Type u_3) (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [CommSemiring R] [AddCommMonoid M] [Module R M] (m : M) : (TrivSqZeroExt.inrCLM R M) m = TrivSqZeroExt.inr m

instance TrivSqZeroExt.instContinuousAdd {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Add R] [Add M] [ContinuousAdd R] [ContinuousAdd M] : ContinuousAdd (TrivSqZeroExt R M)

instance TrivSqZeroExt.instContinuousMulOfContinuousSMulMulOppositeOfContinuousAdd {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Mul R] [Add M] [SMul R M] [SMul Rᵐᵒᵖ M] [ContinuousMul R] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] [ContinuousAdd M] : ContinuousMul (TrivSqZeroExt R M)

instance TrivSqZeroExt.instContinuousNeg {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Neg R] [Neg M] [ContinuousNeg R] [ContinuousNeg M] : ContinuousNeg (TrivSqZeroExt R M)

theorem TrivSqZeroExt.topologicalSemiring {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] [TopologicalSemiring R] [ContinuousAdd M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] : TopologicalSemiring (TrivSqZeroExt R M)

This is not an instance due to complaints by the fails_quickly linter. At any rate, we only really care about the TopologicalRing instance below.

instance TrivSqZeroExt.instTopologicalRingOfTopologicalAddGroupOfContinuousSMulMulOpposite {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Ring R] [AddCommGroup M] [Module R M] [Module Rᵐᵒᵖ M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] : TopologicalRing (TrivSqZeroExt R M)

instance TrivSqZeroExt.instContinuousConstSMul {S : Type u_2} {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [SMul S R] [SMul S M] [ContinuousConstSMul S R] [ContinuousConstSMul S M] : ContinuousConstSMul S (TrivSqZeroExt R M)

instance TrivSqZeroExt.instContinuousSMul {S : Type u_2} {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [TopologicalSpace S] [SMul S R] [SMul S M] [ContinuousSMul S R] [ContinuousSMul S M] : ContinuousSMul S (TrivSqZeroExt R M)

theorem TrivSqZeroExt.hasSum_inl {α : Type u_1} {R : Type u_3} (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [AddCommMonoid R] [AddCommMonoid M] {f : α → R} {a : R} (h : HasSum f a) : HasSum (fun (x : α) => TrivSqZeroExt.inl (f x)) (TrivSqZeroExt.inl a)

theorem TrivSqZeroExt.hasSum_inr {α : Type u_1} {R : Type u_3} (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [AddCommMonoid R] [AddCommMonoid M] {f : α → M} {a : M} (h : HasSum f a) : HasSum (fun (x : α) => TrivSqZeroExt.inr (f x)) (TrivSqZeroExt.inr a)

theorem TrivSqZeroExt.hasSum_fst {α : Type u_1} {R : Type u_3} (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [AddCommMonoid R] [AddCommMonoid M] {f : α → TrivSqZeroExt R M} {a : TrivSqZeroExt R M} (h : HasSum f a) : HasSum (fun (x : α) => (f x).fst) a.fst

theorem TrivSqZeroExt.hasSum_snd {α : Type u_1} {R : Type u_3} (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [AddCommMonoid R] [AddCommMonoid M] {f : α → TrivSqZeroExt R M} {a : TrivSqZeroExt R M} (h : HasSum f a) : HasSum (fun (x : α) => (f x).snd) a.snd

instance TrivSqZeroExt.instUniformSpace {R : Type u_3} {M : Type u_4} [UniformSpace R] [UniformSpace M] : UniformSpace (TrivSqZeroExt R M)

Equations

• TrivSqZeroExt.instUniformSpace = UniformSpace.mk UniformSpace.uniformity ⋯ ⋯ ⋯

instance TrivSqZeroExt.instCompleteSpace {R : Type u_3} {M : Type u_4} [UniformSpace R] [UniformSpace M] [CompleteSpace R] [CompleteSpace M] : CompleteSpace (TrivSqZeroExt R M)

instance TrivSqZeroExt.instUniformAddGroup {R : Type u_3} {M : Type u_4} [UniformSpace R] [UniformSpace M] [AddGroup R] [AddGroup M] [UniformAddGroup R] [UniformAddGroup M] : UniformAddGroup (TrivSqZeroExt R M)

theorem TrivSqZeroExt.uniformity_def {R : Type u_3} {M : Type u_4} [UniformSpace R] [UniformSpace M] : uniformity (TrivSqZeroExt R M) = Filter.comap (fun (p : TrivSqZeroExt R M × TrivSqZeroExt R M) => (p.1.fst, p.2.fst)) (uniformity R) ⊓ Filter.comap (fun (p : TrivSqZeroExt R M × TrivSqZeroExt R M) => (p.1.snd, p.2.snd)) (uniformity M)

theorem TrivSqZeroExt.uniformContinuous_fst {R : Type u_3} {M : Type u_4} [UniformSpace R] [UniformSpace M] : UniformContinuous TrivSqZeroExt.fst

theorem TrivSqZeroExt.uniformContinuous_snd {R : Type u_3} {M : Type u_4} [UniformSpace R] [UniformSpace M] : UniformContinuous TrivSqZeroExt.snd

theorem TrivSqZeroExt.uniformContinuous_inl {R : Type u_3} {M : Type u_4} [UniformSpace R] [UniformSpace M] [Zero M] : UniformContinuous TrivSqZeroExt.inl

theorem TrivSqZeroExt.uniformContinuous_inr {R : Type u_3} {M : Type u_4} [UniformSpace R] [UniformSpace M] [Zero R] : UniformContinuous TrivSqZeroExt.inr