Open subgroups of a topological group

This files builds the lattice OpenSubgroup G of open subgroups in a topological group G, and its additive version OpenAddSubgroup. This lattice has a top element, the subgroup of all elements, but no bottom element in general. The trivial subgroup which is the natural candidate bottom has no reason to be open (this happens only in discrete groups).

Note that this notion is especially relevant in a non-archimedean context, for instance for p-adic groups.

Main declarations

• OpenSubgroup.isClosed: An open subgroup is automatically closed.
• Subgroup.isOpen_mono: A subgroup containing an open subgroup is open. There are also versions for additive groups, submodules and ideals.
• OpenSubgroup.comap: Open subgroups can be pulled back by a continuous group morphism.

TODO

• Prove that the identity component of a locally path connected group is an open subgroup. Up to now this file is really geared towards non-archimedean algebra, not Lie groups.

structure OpenAddSubgroup (G : Type u_1) [AddGroup G] [TopologicalSpace G] extends AddSubgroup G : Type u_1

The type of open subgroups of a topological additive group.

• carrier : Set G
• add_mem' {a b : G} : a ∈ (↑self).carrier → b ∈ (↑self).carrier → a + b ∈ (↑self).carrier
• zero_mem' : 0 ∈ (↑self).carrier
• neg_mem' {x : G} : x ∈ (↑self).carrier → -x ∈ (↑self).carrier
• isOpen' : IsOpen (↑self).carrier

structure OpenSubgroup (G : Type u_1) [Group G] [TopologicalSpace G] extends Subgroup G : Type u_1

The type of open subgroups of a topological group.

• carrier : Set G
• mul_mem' {a b : G} : a ∈ (↑self).carrier → b ∈ (↑self).carrier → a * b ∈ (↑self).carrier
• one_mem' : 1 ∈ (↑self).carrier
• inv_mem' {x : G} : x ∈ (↑self).carrier → x⁻¹ ∈ (↑self).carrier
• isOpen' : IsOpen (↑self).carrier

instance OpenSubgroup.hasCoeSubgroup {G : Type u_1} [Group G] [TopologicalSpace G] : CoeTC (OpenSubgroup G) (Subgroup G)

Equations

• OpenSubgroup.hasCoeSubgroup = { coe := OpenSubgroup.toSubgroup }

instance OpenAddSubgroup.hasCoeAddSubgroup {G : Type u_1} [AddGroup G] [TopologicalSpace G] : CoeTC (OpenAddSubgroup G) (AddSubgroup G)

Equations

• OpenAddSubgroup.hasCoeAddSubgroup = { coe := OpenAddSubgroup.toAddSubgroup }

theorem OpenSubgroup.toSubgroup_injective {G : Type u_1} [Group G] [TopologicalSpace G] : Function.Injective toSubgroup

theorem OpenAddSubgroup.toAddSubgroup_injective {G : Type u_1} [AddGroup G] [TopologicalSpace G] : Function.Injective toAddSubgroup

instance OpenSubgroup.instSetLike {G : Type u_1} [Group G] [TopologicalSpace G] : SetLike (OpenSubgroup G) G

Equations

• OpenSubgroup.instSetLike = { coe := fun (U : OpenSubgroup G) => ↑↑U, coe_injective' := ⋯ }

instance OpenAddSubgroup.instSetLike {G : Type u_1} [AddGroup G] [TopologicalSpace G] : SetLike (OpenAddSubgroup G) G

Equations

• OpenAddSubgroup.instSetLike = { coe := fun (U : OpenAddSubgroup G) => ↑↑U, coe_injective' := ⋯ }

instance OpenSubgroup.instSubgroupClass {G : Type u_1} [Group G] [TopologicalSpace G] : SubgroupClass (OpenSubgroup G) G

instance OpenAddSubgroup.instAddSubgroupClass {G : Type u_1} [AddGroup G] [TopologicalSpace G] : AddSubgroupClass (OpenAddSubgroup G) G

def OpenSubgroup.toOpens {G : Type u_1} [Group G] [TopologicalSpace G] (U : OpenSubgroup G) : TopologicalSpace.Opens G

Coercion from OpenSubgroup G to Opens G.

Equations

• ↑U = { carrier := ↑U, is_open' := ⋯ }

def OpenAddSubgroup.toOpens {G : Type u_1} [AddGroup G] [TopologicalSpace G] (U : OpenAddSubgroup G) : TopologicalSpace.Opens G

Coercion from OpenAddSubgroup G to Opens G.

Equations

• ↑U = { carrier := ↑U, is_open' := ⋯ }

instance OpenSubgroup.hasCoeOpens {G : Type u_1} [Group G] [TopologicalSpace G] : CoeTC (OpenSubgroup G) (TopologicalSpace.Opens G)

Equations

• OpenSubgroup.hasCoeOpens = { coe := OpenSubgroup.toOpens }

instance OpenAddSubgroup.hasCoeOpens {G : Type u_1} [AddGroup G] [TopologicalSpace G] : CoeTC (OpenAddSubgroup G) (TopologicalSpace.Opens G)

Equations

• OpenAddSubgroup.hasCoeOpens = { coe := OpenAddSubgroup.toOpens }

@[simp]

theorem OpenSubgroup.coe_toOpens {G : Type u_1} [Group G] [TopologicalSpace G] {U : OpenSubgroup G} : ↑↑U = ↑U

@[simp]

theorem OpenAddSubgroup.coe_toOpens {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U : OpenAddSubgroup G} : ↑↑U = ↑U

@[simp]

theorem OpenSubgroup.coe_toSubgroup {G : Type u_1} [Group G] [TopologicalSpace G] {U : OpenSubgroup G} : ↑↑U = ↑U

@[simp]

theorem OpenAddSubgroup.coe_toAddSubgroup {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U : OpenAddSubgroup G} : ↑↑U = ↑U

@[simp]

theorem OpenSubgroup.mem_toOpens {G : Type u_1} [Group G] [TopologicalSpace G] {U : OpenSubgroup G} {g : G} : g ∈ ↑U ↔ g ∈ U

@[simp]

theorem OpenAddSubgroup.mem_toOpens {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U : OpenAddSubgroup G} {g : G} : g ∈ ↑U ↔ g ∈ U

@[simp]

theorem OpenSubgroup.mem_toSubgroup {G : Type u_1} [Group G] [TopologicalSpace G] {U : OpenSubgroup G} {g : G} : g ∈ ↑U ↔ g ∈ U

@[simp]

theorem OpenAddSubgroup.mem_toAddSubgroup {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U : OpenAddSubgroup G} {g : G} : g ∈ ↑U ↔ g ∈ U

theorem OpenSubgroup.ext {G : Type u_1} [Group G] [TopologicalSpace G] {U V : OpenSubgroup G} (h : ∀ (x : G), x ∈ U ↔ x ∈ V) : U = V

theorem OpenAddSubgroup.ext {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U V : OpenAddSubgroup G} (h : ∀ (x : G), x ∈ U ↔ x ∈ V) : U = V

theorem OpenAddSubgroup.ext_iff {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U V : OpenAddSubgroup G} : U = V ↔ ∀ (x : G), x ∈ U ↔ x ∈ V

theorem OpenSubgroup.ext_iff {G : Type u_1} [Group G] [TopologicalSpace G] {U V : OpenSubgroup G} : U = V ↔ ∀ (x : G), x ∈ U ↔ x ∈ V

theorem OpenSubgroup.isOpen {G : Type u_1} [Group G] [TopologicalSpace G] (U : OpenSubgroup G) : IsOpen ↑U

theorem OpenAddSubgroup.isOpen {G : Type u_1} [AddGroup G] [TopologicalSpace G] (U : OpenAddSubgroup G) : IsOpen ↑U

theorem OpenSubgroup.mem_nhds_one {G : Type u_1} [Group G] [TopologicalSpace G] (U : OpenSubgroup G) : ↑U ∈ nhds 1

theorem OpenAddSubgroup.mem_nhds_zero {G : Type u_1} [AddGroup G] [TopologicalSpace G] (U : OpenAddSubgroup G) : ↑U ∈ nhds 0

instance OpenSubgroup.instTop {G : Type u_1} [Group G] [TopologicalSpace G] : Top (OpenSubgroup G)

Equations

• OpenSubgroup.instTop = { top := { toSubgroup := ⊤, isOpen' := ⋯ } }

instance OpenAddSubgroup.instTop {G : Type u_1} [AddGroup G] [TopologicalSpace G] : Top (OpenAddSubgroup G)

Equations

• OpenAddSubgroup.instTop = { top := { toAddSubgroup := ⊤, isOpen' := ⋯ } }

@[simp]

theorem OpenSubgroup.mem_top {G : Type u_1} [Group G] [TopologicalSpace G] (x : G) : x ∈ ⊤

@[simp]

theorem OpenAddSubgroup.mem_top {G : Type u_1} [AddGroup G] [TopologicalSpace G] (x : G) : x ∈ ⊤

@[simp]

theorem OpenSubgroup.coe_top {G : Type u_1} [Group G] [TopologicalSpace G] : ↑⊤ = Set.univ

@[simp]

theorem OpenAddSubgroup.coe_top {G : Type u_1} [AddGroup G] [TopologicalSpace G] : ↑⊤ = Set.univ

@[simp]

theorem OpenSubgroup.toSubgroup_top {G : Type u_1} [Group G] [TopologicalSpace G] : ↑⊤ = ⊤

@[simp]

theorem OpenAddSubgroup.toAddSubgroup_top {G : Type u_1} [AddGroup G] [TopologicalSpace G] : ↑⊤ = ⊤

@[simp]

theorem OpenSubgroup.toOpens_top {G : Type u_1} [Group G] [TopologicalSpace G] : ↑⊤ = ⊤

@[simp]

theorem OpenAddSubgroup.toOpens_top {G : Type u_1} [AddGroup G] [TopologicalSpace G] : ↑⊤ = ⊤

instance OpenSubgroup.instInhabited {G : Type u_1} [Group G] [TopologicalSpace G] : Inhabited (OpenSubgroup G)

Equations

• OpenSubgroup.instInhabited = { default := ⊤ }

instance OpenAddSubgroup.instInhabited {G : Type u_1} [AddGroup G] [TopologicalSpace G] : Inhabited (OpenAddSubgroup G)

Equations

• OpenAddSubgroup.instInhabited = { default := ⊤ }

theorem OpenSubgroup.isClosed {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousMul G] (U : OpenSubgroup G) : IsClosed ↑U

theorem OpenAddSubgroup.isClosed {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] (U : OpenAddSubgroup G) : IsClosed ↑U

theorem OpenSubgroup.isClopen {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousMul G] (U : OpenSubgroup G) : IsClopen ↑U

theorem OpenAddSubgroup.isClopen {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] (U : OpenAddSubgroup G) : IsClopen ↑U

def OpenSubgroup.prod {G : Type u_1} [Group G] [TopologicalSpace G] {H : Type u_2} [Group H] [TopologicalSpace H] (U : OpenSubgroup G) (V : OpenSubgroup H) : OpenSubgroup (G × H)

The product of two open subgroups as an open subgroup of the product group.

Equations

• U.prod V = { toSubgroup := (↑U).prod ↑V, isOpen' := ⋯ }

def OpenAddSubgroup.prod {G : Type u_1} [AddGroup G] [TopologicalSpace G] {H : Type u_2} [AddGroup H] [TopologicalSpace H] (U : OpenAddSubgroup G) (V : OpenAddSubgroup H) : OpenAddSubgroup (G × H)

The product of two open subgroups as an open subgroup of the product group.

Equations

• U.prod V = { toAddSubgroup := (↑U).prod ↑V, isOpen' := ⋯ }

@[deprecated OpenAddSubgroup.prod (since := "2025-03-11")]

def OpenAddSubgroup.sum {G : Type u_1} [AddGroup G] [TopologicalSpace G] {H : Type u_2} [AddGroup H] [TopologicalSpace H] (U : OpenAddSubgroup G) (V : OpenAddSubgroup H) : OpenAddSubgroup (G × H)

Alias of OpenAddSubgroup.prod.

---

The product of two open subgroups as an open subgroup of the product group.

Equations

• @OpenAddSubgroup.sum = @OpenAddSubgroup.prod

@[simp]

theorem OpenSubgroup.coe_prod {G : Type u_1} [Group G] [TopologicalSpace G] {H : Type u_2} [Group H] [TopologicalSpace H] (U : OpenSubgroup G) (V : OpenSubgroup H) : ↑(U.prod V) = ↑U ×ˢ ↑V

@[simp]

theorem OpenAddSubgroup.coe_prod {G : Type u_1} [AddGroup G] [TopologicalSpace G] {H : Type u_2} [AddGroup H] [TopologicalSpace H] (U : OpenAddSubgroup G) (V : OpenAddSubgroup H) : ↑(U.prod V) = ↑U ×ˢ ↑V

@[deprecated OpenAddSubgroup.coe_prod (since := "2025-03-11")]

theorem OpenAddSubgroup.coe_sum {G : Type u_1} [AddGroup G] [TopologicalSpace G] {H : Type u_2} [AddGroup H] [TopologicalSpace H] (U : OpenAddSubgroup G) (V : OpenAddSubgroup H) : ↑(U.prod V) = ↑U ×ˢ ↑V

Alias of OpenAddSubgroup.coe_prod.

@[simp]

theorem OpenSubgroup.toSubgroup_prod {G : Type u_1} [Group G] [TopologicalSpace G] {H : Type u_2} [Group H] [TopologicalSpace H] (U : OpenSubgroup G) (V : OpenSubgroup H) : ↑(U.prod V) = (↑U).prod ↑V

@[simp]

theorem OpenAddSubgroup.toAddSubgroup_prod {G : Type u_1} [AddGroup G] [TopologicalSpace G] {H : Type u_2} [AddGroup H] [TopologicalSpace H] (U : OpenAddSubgroup G) (V : OpenAddSubgroup H) : ↑(U.prod V) = (↑U).prod ↑V

@[deprecated OpenAddSubgroup.toAddSubgroup_prod (since := "2025-03-11")]

theorem OpenAddSubgroup.toAddSubgroup_sum {G : Type u_1} [AddGroup G] [TopologicalSpace G] {H : Type u_2} [AddGroup H] [TopologicalSpace H] (U : OpenAddSubgroup G) (V : OpenAddSubgroup H) : ↑(U.prod V) = (↑U).prod ↑V

Alias of OpenAddSubgroup.toAddSubgroup_prod.

instance OpenSubgroup.instInfOpenSubgroup {G : Type u_1} [Group G] [TopologicalSpace G] : Min (OpenSubgroup G)

Equations

• OpenSubgroup.instInfOpenSubgroup = { min := fun (U V : OpenSubgroup G) => { toSubgroup := ↑U ⊓ ↑V, isOpen' := ⋯ } }

instance OpenAddSubgroup.instInfOpenAddSubgroup {G : Type u_1} [AddGroup G] [TopologicalSpace G] : Min (OpenAddSubgroup G)

Equations

• OpenAddSubgroup.instInfOpenAddSubgroup = { min := fun (U V : OpenAddSubgroup G) => { toAddSubgroup := ↑U ⊓ ↑V, isOpen' := ⋯ } }

@[simp]

theorem OpenSubgroup.coe_inf {G : Type u_1} [Group G] [TopologicalSpace G] {U V : OpenSubgroup G} : ↑(U ⊓ V) = ↑U ∩ ↑V

@[simp]

theorem OpenAddSubgroup.coe_inf {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U V : OpenAddSubgroup G} : ↑(U ⊓ V) = ↑U ∩ ↑V

@[simp]

theorem OpenSubgroup.toSubgroup_inf {G : Type u_1} [Group G] [TopologicalSpace G] {U V : OpenSubgroup G} : ↑(U ⊓ V) = ↑U ⊓ ↑V

@[simp]

theorem OpenAddSubgroup.toAddSubgroup_inf {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U V : OpenAddSubgroup G} : ↑(U ⊓ V) = ↑U ⊓ ↑V

@[simp]

theorem OpenSubgroup.toOpens_inf {G : Type u_1} [Group G] [TopologicalSpace G] {U V : OpenSubgroup G} : ↑(U ⊓ V) = ↑U ⊓ ↑V

@[simp]

theorem OpenAddSubgroup.toOpens_inf {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U V : OpenAddSubgroup G} : ↑(U ⊓ V) = ↑U ⊓ ↑V

@[simp]

theorem OpenSubgroup.mem_inf {G : Type u_1} [Group G] [TopologicalSpace G] {U V : OpenSubgroup G} {x : G} : x ∈ U ⊓ V ↔ x ∈ U ∧ x ∈ V

@[simp]

theorem OpenAddSubgroup.mem_inf {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U V : OpenAddSubgroup G} {x : G} : x ∈ U ⊓ V ↔ x ∈ U ∧ x ∈ V

instance OpenSubgroup.instPartialOrderOpenSubgroup {G : Type u_1} [Group G] [TopologicalSpace G] : PartialOrder (OpenSubgroup G)

Equations

• OpenSubgroup.instPartialOrderOpenSubgroup = inferInstance

instance OpenAddSubgroup.instPartialOrderOpenAddSubgroup {G : Type u_1} [AddGroup G] [TopologicalSpace G] : PartialOrder (OpenAddSubgroup G)

Equations

• OpenAddSubgroup.instPartialOrderOpenAddSubgroup = inferInstance

instance OpenSubgroup.instSemilatticeInfOpenSubgroup {G : Type u_1} [Group G] [TopologicalSpace G] : SemilatticeInf (OpenSubgroup G)

Equations

• OpenSubgroup.instSemilatticeInfOpenSubgroup = { toPartialOrder := OpenSubgroup.instPartialOrderOpenSubgroup, inf := SemilatticeInf.inf, inf_le_left := ⋯, inf_le_right := ⋯, le_inf := ⋯ }

instance OpenAddSubgroup.instSemilatticeInfOpenAddSubgroup {G : Type u_1} [AddGroup G] [TopologicalSpace G] : SemilatticeInf (OpenAddSubgroup G)

Equations

• One or more equations did not get rendered due to their size.

instance OpenSubgroup.instOrderTop {G : Type u_1} [Group G] [TopologicalSpace G] : OrderTop (OpenSubgroup G)

Equations

• OpenSubgroup.instOrderTop = { toTop := OpenSubgroup.instTop, le_top := ⋯ }

instance OpenAddSubgroup.instOrderTop {G : Type u_1} [AddGroup G] [TopologicalSpace G] : OrderTop (OpenAddSubgroup G)

Equations

• OpenAddSubgroup.instOrderTop = { toTop := OpenAddSubgroup.instTop, le_top := ⋯ }

@[simp]

theorem OpenSubgroup.toSubgroup_le {G : Type u_1} [Group G] [TopologicalSpace G] {U V : OpenSubgroup G} : ↑U ≤ ↑V ↔ U ≤ V

@[simp]

theorem OpenAddSubgroup.toAddSubgroup_le {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U V : OpenAddSubgroup G} : ↑U ≤ ↑V ↔ U ≤ V

def OpenSubgroup.comap {G : Type u_1} [Group G] [TopologicalSpace G] {N : Type u_2} [Group N] [TopologicalSpace N] (f : G →* N) (hf : Continuous ⇑f) (H : OpenSubgroup N) : OpenSubgroup G

The preimage of an OpenSubgroup along a continuous Monoid homomorphism is an OpenSubgroup.

Equations

• OpenSubgroup.comap f hf H = { toSubgroup := Subgroup.comap f ↑H, isOpen' := ⋯ }

def OpenAddSubgroup.comap {G : Type u_1} [AddGroup G] [TopologicalSpace G] {N : Type u_2} [AddGroup N] [TopologicalSpace N] (f : G →+ N) (hf : Continuous ⇑f) (H : OpenAddSubgroup N) : OpenAddSubgroup G

The preimage of an OpenAddSubgroup along a continuous AddMonoid homomorphism is an OpenAddSubgroup.

Equations

• OpenAddSubgroup.comap f hf H = { toAddSubgroup := AddSubgroup.comap f ↑H, isOpen' := ⋯ }

@[simp]

theorem OpenSubgroup.coe_comap {G : Type u_1} [Group G] [TopologicalSpace G] {N : Type u_2} [Group N] [TopologicalSpace N] (H : OpenSubgroup N) (f : G →* N) (hf : Continuous ⇑f) : ↑(comap f hf H) = ⇑f ⁻¹' ↑H

@[simp]

theorem OpenAddSubgroup.coe_comap {G : Type u_1} [AddGroup G] [TopologicalSpace G] {N : Type u_2} [AddGroup N] [TopologicalSpace N] (H : OpenAddSubgroup N) (f : G →+ N) (hf : Continuous ⇑f) : ↑(comap f hf H) = ⇑f ⁻¹' ↑H

@[simp]

theorem OpenSubgroup.toSubgroup_comap {G : Type u_1} [Group G] [TopologicalSpace G] {N : Type u_2} [Group N] [TopologicalSpace N] (H : OpenSubgroup N) (f : G →* N) (hf : Continuous ⇑f) : ↑(comap f hf H) = Subgroup.comap f ↑H

@[simp]

theorem OpenAddSubgroup.toAddSubgroup_comap {G : Type u_1} [AddGroup G] [TopologicalSpace G] {N : Type u_2} [AddGroup N] [TopologicalSpace N] (H : OpenAddSubgroup N) (f : G →+ N) (hf : Continuous ⇑f) : ↑(comap f hf H) = AddSubgroup.comap f ↑H

@[simp]

theorem OpenSubgroup.mem_comap {G : Type u_1} [Group G] [TopologicalSpace G] {N : Type u_2} [Group N] [TopologicalSpace N] {H : OpenSubgroup N} {f : G →* N} {hf : Continuous ⇑f} {x : G} : x ∈ comap f hf H ↔ f x ∈ H

@[simp]

theorem OpenAddSubgroup.mem_comap {G : Type u_1} [AddGroup G] [TopologicalSpace G] {N : Type u_2} [AddGroup N] [TopologicalSpace N] {H : OpenAddSubgroup N} {f : G →+ N} {hf : Continuous ⇑f} {x : G} : x ∈ comap f hf H ↔ f x ∈ H

theorem OpenSubgroup.comap_comap {G : Type u_1} [Group G] [TopologicalSpace G] {N : Type u_2} [Group N] [TopologicalSpace N] {P : Type u_3} [Group P] [TopologicalSpace P] (K : OpenSubgroup P) (f₂ : N →* P) (hf₂ : Continuous ⇑f₂) (f₁ : G →* N) (hf₁ : Continuous ⇑f₁) : comap f₁ hf₁ (comap f₂ hf₂ K) = comap (f₂.comp f₁) ⋯ K

theorem OpenAddSubgroup.comap_comap {G : Type u_1} [AddGroup G] [TopologicalSpace G] {N : Type u_2} [AddGroup N] [TopologicalSpace N] {P : Type u_3} [AddGroup P] [TopologicalSpace P] (K : OpenAddSubgroup P) (f₂ : N →+ P) (hf₂ : Continuous ⇑f₂) (f₁ : G →+ N) (hf₁ : Continuous ⇑f₁) : comap f₁ hf₁ (comap f₂ hf₂ K) = comap (f₂.comp f₁) ⋯ K

theorem Subgroup.isOpen_of_mem_nhds {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousMul G] (H : Subgroup G) {g : G} (hg : ↑H ∈ nhds g) : IsOpen ↑H

theorem AddSubgroup.isOpen_of_mem_nhds {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] (H : AddSubgroup G) {g : G} (hg : ↑H ∈ nhds g) : IsOpen ↑H

theorem Subgroup.isOpen_mono {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousMul G] {H₁ H₂ : Subgroup G} (h : H₁ ≤ H₂) (h₁ : IsOpen ↑H₁) : IsOpen ↑H₂

theorem AddSubgroup.isOpen_mono {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] {H₁ H₂ : AddSubgroup G} (h : H₁ ≤ H₂) (h₁ : IsOpen ↑H₁) : IsOpen ↑H₂

theorem Subgroup.isOpen_of_openSubgroup {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousMul G] (H : Subgroup G) {U : OpenSubgroup G} (h : ↑U ≤ H) : IsOpen ↑H

theorem AddSubgroup.isOpen_of_openAddSubgroup {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] (H : AddSubgroup G) {U : OpenAddSubgroup G} (h : ↑U ≤ H) : IsOpen ↑H

theorem Subgroup.isOpen_of_one_mem_interior {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousMul G] (H : Subgroup G) (h_1_int : 1 ∈ interior ↑H) : IsOpen ↑H

If a subgroup of a topological group has 1 in its interior, then it is open.

theorem AddSubgroup.isOpen_of_zero_mem_interior {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] (H : AddSubgroup G) (h_1_int : 0 ∈ interior ↑H) : IsOpen ↑H

If a subgroup of an additive topological group has 0 in its interior, then it is open.

theorem Subgroup.isClosed_of_isOpen {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousMul G] (U : Subgroup G) (h : IsOpen ↑U) : IsClosed ↑U

theorem AddSubgroup.isClosed_of_isOpen {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] (U : AddSubgroup G) (h : IsOpen ↑U) : IsClosed ↑U

theorem Subgroup.subgroupOf_isOpen {G : Type u_1} [Group G] [TopologicalSpace G] (U K : Subgroup G) (h : IsOpen ↑K) : IsOpen ↑(K.subgroupOf U)

theorem AddSubgroup.addSubgroupOf_isOpen {G : Type u_1} [AddGroup G] [TopologicalSpace G] (U K : AddSubgroup G) (h : IsOpen ↑K) : IsOpen ↑(K.addSubgroupOf U)

theorem Subgroup.discreteTopology {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousMul G] (U : Subgroup G) (h : IsOpen ↑U) : DiscreteTopology (G ⧸ U)

theorem AddSubgroup.discreteTopology {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] (U : AddSubgroup G) (h : IsOpen ↑U) : DiscreteTopology (G ⧸ U)

instance Subgroup.instDiscreteTopologyQuotientOfContinuousMul {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousMul G] (U : OpenSubgroup G) : DiscreteTopology (G ⧸ ↑U)

instance AddSubgroup.instDiscreteTopologyQuotientOfContinuousAdd {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] (U : OpenAddSubgroup G) : DiscreteTopology (G ⧸ ↑U)

theorem Subgroup.quotient_finite_of_isOpen {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousMul G] [CompactSpace G] (U : Subgroup G) (h : IsOpen ↑U) : Finite (G ⧸ U)

theorem AddSubgroup.quotient_finite_of_isOpen {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] [CompactSpace G] (U : AddSubgroup G) (h : IsOpen ↑U) : Finite (G ⧸ U)

instance Subgroup.instFiniteQuotientOfContinuousMulOfCompactSpace {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousMul G] [CompactSpace G] (U : OpenSubgroup G) : Finite (G ⧸ ↑U)

instance AddSubgroup.instFiniteQuotientOfContinuousAddOfCompactSpace {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] [CompactSpace G] (U : OpenAddSubgroup G) : Finite (G ⧸ ↑U)

theorem Subgroup.quotient_finite_of_isOpen' {G : Type u_1} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] (U : Subgroup G) (K : Subgroup ↥U) (hUopen : IsOpen ↑U) (hKopen : IsOpen ↑K) : Finite (↥U ⧸ K)

theorem AddSubgroup.quotient_finite_of_isOpen' {G : Type u_1} [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] [CompactSpace G] (U : AddSubgroup G) (K : AddSubgroup ↥U) (hUopen : IsOpen ↑U) (hKopen : IsOpen ↑K) : Finite (↥U ⧸ K)

instance Subgroup.instFiniteQuotientSubtypeMemOpenSubgroupOfIsTopologicalGroupOfCompactSpace {G : Type u_1} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] (U : OpenSubgroup G) (K : OpenSubgroup ↥U) : Finite (↥U ⧸ ↑K)

instance AddSubgroup.instFiniteQuotientSubtypeMemOpenAddSubgroupOfIsTopologicalAddGroupOfCompactSpace {G : Type u_1} [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] [CompactSpace G] (U : OpenAddSubgroup G) (K : OpenAddSubgroup ↥U) : Finite (↥U ⧸ ↑K)

instance OpenSubgroup.instMax {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousMul G] : Max (OpenSubgroup G)

Equations

• OpenSubgroup.instMax = { max := fun (U V : OpenSubgroup G) => { toSubgroup := ↑U ⊔ ↑V, isOpen' := ⋯ } }

instance OpenAddSubgroup.instMax {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] : Max (OpenAddSubgroup G)

Equations

• OpenAddSubgroup.instMax = { max := fun (U V : OpenAddSubgroup G) => { toAddSubgroup := ↑U ⊔ ↑V, isOpen' := ⋯ } }

@[simp]

theorem OpenSubgroup.toSubgroup_sup {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousMul G] (U V : OpenSubgroup G) : ↑(U ⊔ V) = ↑U ⊔ ↑V

@[simp]

theorem OpenAddSubgroup.toAddSubgroup_sup {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] (U V : OpenAddSubgroup G) : ↑(U ⊔ V) = ↑U ⊔ ↑V

instance OpenSubgroup.instLattice {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousMul G] : Lattice (OpenSubgroup G)

Equations

• One or more equations did not get rendered due to their size.

instance OpenAddSubgroup.instLattice {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] : Lattice (OpenAddSubgroup G)

Equations

• One or more equations did not get rendered due to their size.

theorem Submodule.isOpen_mono {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [TopologicalSpace M] [IsTopologicalAddGroup M] [Module R M] {U P : Submodule R M} (h : U ≤ P) (hU : IsOpen ↑U) : IsOpen ↑P

theorem Ideal.isOpen_of_isOpen_subideal {R : Type u_1} [CommRing R] [TopologicalSpace R] [IsTopologicalRing R] {U I : Ideal R} (h : U ≤ I) (hU : IsOpen ↑U) : IsOpen ↑I

Open normal subgroups of a topological group

This section builds the lattice OpenNormalSubgroup G of open subgroups in a topological group G, and its additive version OpenNormalAddSubgroup.

structure OpenNormalSubgroup (G : Type u) [Group G] [TopologicalSpace G] extends OpenSubgroup G : Type u

The type of open normal subgroups of a topological group.

• carrier : Set G
• mul_mem' {a b : G} : a ∈ (↑self.toOpenSubgroup).carrier → b ∈ (↑self.toOpenSubgroup).carrier → a * b ∈ (↑self.toOpenSubgroup).carrier
• one_mem' : 1 ∈ (↑self.toOpenSubgroup).carrier
• inv_mem' {x : G} : x ∈ (↑self.toOpenSubgroup).carrier → x⁻¹ ∈ (↑self.toOpenSubgroup).carrier
• isOpen' : IsOpen (↑self.toOpenSubgroup).carrier
• isNormal' : (↑self.toOpenSubgroup).Normal

theorem OpenNormalSubgroup.ext_iff {G : Type u} {inst✝ : Group G} {inst✝¹ : TopologicalSpace G} {x y : OpenNormalSubgroup G} : x = y ↔ (↑x.toOpenSubgroup).carrier = (↑y.toOpenSubgroup).carrier

theorem OpenNormalSubgroup.ext {G : Type u} {inst✝ : Group G} {inst✝¹ : TopologicalSpace G} {x y : OpenNormalSubgroup G} (carrier : (↑x.toOpenSubgroup).carrier = (↑y.toOpenSubgroup).carrier) : x = y

structure OpenNormalAddSubgroup (G : Type u) [AddGroup G] [TopologicalSpace G] extends OpenAddSubgroup G : Type u

The type of open normal subgroups of a topological additive group.

• carrier : Set G
• add_mem' {a b : G} : a ∈ (↑self.toOpenAddSubgroup).carrier → b ∈ (↑self.toOpenAddSubgroup).carrier → a + b ∈ (↑self.toOpenAddSubgroup).carrier
• zero_mem' : 0 ∈ (↑self.toOpenAddSubgroup).carrier
• neg_mem' {x : G} : x ∈ (↑self.toOpenAddSubgroup).carrier → -x ∈ (↑self.toOpenAddSubgroup).carrier
• isOpen' : IsOpen (↑self.toOpenAddSubgroup).carrier
• isNormal' : (↑self.toOpenAddSubgroup).Normal

theorem OpenNormalAddSubgroup.ext {G : Type u} {inst✝ : AddGroup G} {inst✝¹ : TopologicalSpace G} {x y : OpenNormalAddSubgroup G} (carrier : (↑x.toOpenAddSubgroup).carrier = (↑y.toOpenAddSubgroup).carrier) : x = y

theorem OpenNormalAddSubgroup.ext_iff {G : Type u} {inst✝ : AddGroup G} {inst✝¹ : TopologicalSpace G} {x y : OpenNormalAddSubgroup G} : x = y ↔ (↑x.toOpenAddSubgroup).carrier = (↑y.toOpenAddSubgroup).carrier

instance OpenNormalSubgroup.instNormal {G : Type u} [Group G] [TopologicalSpace G] (H : OpenNormalSubgroup G) : (↑H.toOpenSubgroup).Normal

instance OpenNormalAddSubgroup.instNormal {G : Type u} [AddGroup G] [TopologicalSpace G] (H : OpenNormalAddSubgroup G) : (↑H.toOpenAddSubgroup).Normal

theorem OpenNormalSubgroup.toSubgroup_injective {G : Type u} [Group G] [TopologicalSpace G] : Function.Injective fun (H : OpenNormalSubgroup G) => ↑H.toOpenSubgroup

theorem OpenNormalAddSubgroup.toAddSubgroup_injective {G : Type u} [AddGroup G] [TopologicalSpace G] : Function.Injective fun (H : OpenNormalAddSubgroup G) => ↑H.toOpenAddSubgroup

instance OpenNormalSubgroup.instSetLike {G : Type u} [Group G] [TopologicalSpace G] : SetLike (OpenNormalSubgroup G) G

Equations

• OpenNormalSubgroup.instSetLike = { coe := fun (U : OpenNormalSubgroup G) => ↑U.toOpenSubgroup, coe_injective' := ⋯ }

instance OpenNormalAddSubgroup.instSetLike {G : Type u} [AddGroup G] [TopologicalSpace G] : SetLike (OpenNormalAddSubgroup G) G

Equations

• OpenNormalAddSubgroup.instSetLike = { coe := fun (U : OpenNormalAddSubgroup G) => ↑U.toOpenAddSubgroup, coe_injective' := ⋯ }

instance OpenNormalSubgroup.instSubgroupClass {G : Type u} [Group G] [TopologicalSpace G] : SubgroupClass (OpenNormalSubgroup G) G

instance OpenNormalAddSubgroup.instAddSubgroupClass {G : Type u} [AddGroup G] [TopologicalSpace G] : AddSubgroupClass (OpenNormalAddSubgroup G) G

instance OpenNormalSubgroup.instCoeSubgroup {G : Type u} [Group G] [TopologicalSpace G] : Coe (OpenNormalSubgroup G) (Subgroup G)

Equations

• OpenNormalSubgroup.instCoeSubgroup = { coe := fun (H : OpenNormalSubgroup G) => ↑H.toOpenSubgroup }

instance OpenNormalAddSubgroup.instCoeAddSubgroup {G : Type u} [AddGroup G] [TopologicalSpace G] : Coe (OpenNormalAddSubgroup G) (AddSubgroup G)

Equations

• OpenNormalAddSubgroup.instCoeAddSubgroup = { coe := fun (H : OpenNormalAddSubgroup G) => ↑H.toOpenAddSubgroup }

instance OpenNormalSubgroup.instPartialOrderOpenNormalSubgroup {G : Type u} [Group G] [TopologicalSpace G] : PartialOrder (OpenNormalSubgroup G)

Equations

• OpenNormalSubgroup.instPartialOrderOpenNormalSubgroup = inferInstance

instance OpenNormalAddSubgroup.instPartialOrderOpenNormalAddSubgroup {G : Type u} [AddGroup G] [TopologicalSpace G] : PartialOrder (OpenNormalAddSubgroup G)

Equations

• OpenNormalAddSubgroup.instPartialOrderOpenNormalAddSubgroup = inferInstance

instance OpenNormalSubgroup.instInfOpenNormalSubgroup {G : Type u} [Group G] [TopologicalSpace G] : Min (OpenNormalSubgroup G)

Equations

• OpenNormalSubgroup.instInfOpenNormalSubgroup = { min := fun (U V : OpenNormalSubgroup G) => { toOpenSubgroup := U.toOpenSubgroup ⊓ V.toOpenSubgroup, isNormal' := ⋯ } }

instance OpenNormalAddSubgroup.instInfOpenNormalAddSubgroup {G : Type u} [AddGroup G] [TopologicalSpace G] : Min (OpenNormalAddSubgroup G)

Equations

• OpenNormalAddSubgroup.instInfOpenNormalAddSubgroup = { min := fun (U V : OpenNormalAddSubgroup G) => { toOpenAddSubgroup := U.toOpenAddSubgroup ⊓ V.toOpenAddSubgroup, isNormal' := ⋯ } }

instance OpenNormalSubgroup.instSemilatticeInfOpenNormalSubgroup {G : Type u} [Group G] [TopologicalSpace G] : SemilatticeInf (OpenNormalSubgroup G)

Equations

• OpenNormalSubgroup.instSemilatticeInfOpenNormalSubgroup = Function.Injective.semilatticeInf SetLike.coe ⋯ ⋯

instance OpenNormalAddSubgroup.instSemilatticeInfOpenNormalAddSubgroup {G : Type u} [AddGroup G] [TopologicalSpace G] : SemilatticeInf (OpenNormalAddSubgroup G)

Equations

• OpenNormalAddSubgroup.instSemilatticeInfOpenNormalAddSubgroup = Function.Injective.semilatticeInf SetLike.coe ⋯ ⋯

instance OpenNormalSubgroup.instMaxOfContinuousMul {G : Type u} [Group G] [TopologicalSpace G] [ContinuousMul G] : Max (OpenNormalSubgroup G)

Equations

• OpenNormalSubgroup.instMaxOfContinuousMul = { max := fun (U V : OpenNormalSubgroup G) => { toOpenSubgroup := U.toOpenSubgroup ⊔ V.toOpenSubgroup, isNormal' := ⋯ } }

instance OpenNormalAddSubgroup.instMaxOfContinuousAdd {G : Type u} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] : Max (OpenNormalAddSubgroup G)

Equations

• OpenNormalAddSubgroup.instMaxOfContinuousAdd = { max := fun (U V : OpenNormalAddSubgroup G) => { toOpenAddSubgroup := U.toOpenAddSubgroup ⊔ V.toOpenAddSubgroup, isNormal' := ⋯ } }

instance OpenNormalSubgroup.instSemilatticeSupOpenNormalSubgroup {G : Type u} [Group G] [TopologicalSpace G] [ContinuousMul G] : SemilatticeSup (OpenNormalSubgroup G)

Equations

• OpenNormalSubgroup.instSemilatticeSupOpenNormalSubgroup = Function.Injective.semilatticeSup (fun (H : OpenNormalSubgroup G) => ↑H.toOpenSubgroup) ⋯ ⋯

instance OpenNormalAddSubgroup.instSemilatticeSupOpenNormalAddSubgroup {G : Type u} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] : SemilatticeSup (OpenNormalAddSubgroup G)

Equations

• OpenNormalAddSubgroup.instSemilatticeSupOpenNormalAddSubgroup = Function.Injective.semilatticeSup (fun (H : OpenNormalAddSubgroup G) => ↑H.toOpenAddSubgroup) ⋯ ⋯

instance OpenNormalSubgroup.instLatticeOfContinuousMul {G : Type u} [Group G] [TopologicalSpace G] [ContinuousMul G] : Lattice (OpenNormalSubgroup G)

Equations

• One or more equations did not get rendered due to their size.

instance OpenNormalAddSubgroup.instLatticeOfContinuousAdd {G : Type u} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] : Lattice (OpenNormalAddSubgroup G)

Equations

• One or more equations did not get rendered due to their size.

Existence of an open subgroup in any clopen neighborhood of the neutral element

This section proves the lemma IsTopologicalGroup.exist_openSubgroup_sub_clopen_nhd_of_one, which states that in a compact topological group, for any clopen neighborhood of 1, there exists an open subgroup contained within it.

structure IsTopologicalAddGroup.addNegClosureNhd {G : Type u_1} [TopologicalSpace G] (T W : Set G) [AddGroup G] : Prop

For a set W, T is a neighborhood of 0 which is open, stable under negation and satisfies T + W ⊆ W.

• nhd : T ∈ nhds 0
• neg : -T = T
• isOpen : IsOpen T
• add : W + T ⊆ W

structure IsTopologicalGroup.mulInvClosureNhd {G : Type u_1} [TopologicalSpace G] (T W : Set G) [Group G] : Prop

For a set W, T is a neighborhood of 1 which is open, statble under inverse and satisfies T * W ⊆ W.

• nhd : T ∈ nhds 1
• inv : T⁻¹ = T
• isOpen : IsOpen T
• mul : W * T ⊆ W

theorem IsTopologicalGroup.exist_mul_closure_nhd {G : Type u_1} [TopologicalSpace G] [Group G] [IsTopologicalGroup G] [CompactSpace G] {W : Set G} (WClopen : IsClopen W) : ∃ T ∈ nhds 1, W * T ⊆ W

theorem IsTopologicalAddGroup.exist_add_closure_nhd {G : Type u_1} [TopologicalSpace G] [AddGroup G] [IsTopologicalAddGroup G] [CompactSpace G] {W : Set G} (WClopen : IsClopen W) : ∃ T ∈ nhds 0, W + T ⊆ W

theorem IsTopologicalGroup.exists_mulInvClosureNhd {G : Type u_1} [TopologicalSpace G] [Group G] [IsTopologicalGroup G] [CompactSpace G] {W : Set G} (WClopen : IsClopen W) : ∃ (T : Set G), mulInvClosureNhd T W

theorem IsTopologicalAddGroup.exists_addNegClosureNhd {G : Type u_1} [TopologicalSpace G] [AddGroup G] [IsTopologicalAddGroup G] [CompactSpace G] {W : Set G} (WClopen : IsClopen W) : ∃ (T : Set G), addNegClosureNhd T W

theorem IsTopologicalGroup.exist_openSubgroup_sub_clopen_nhd_of_one {G : Type u_2} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] {W : Set G} (WClopen : IsClopen W) (einW : 1 ∈ W) : ∃ (H : OpenSubgroup G), ↑H ⊆ W

theorem IsTopologicalAddGroup.exist_openAddSubgroup_sub_clopen_nhd_of_zero {G : Type u_2} [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] [CompactSpace G] {W : Set G} (WClopen : IsClopen W) (einW : 0 ∈ W) : ∃ (H : OpenAddSubgroup G), ↑H ⊆ W