Documentation

Mathlib.Topology.Algebra.OpenSubgroup

Open subgroups of a topological groups #

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 #

TODO #

structure OpenAddSubgroup (G : Type u_1) [AddGroup G] [TopologicalSpace G] extends AddSubgroup :
Type u_1
  • carrier : Set G
  • add_mem' : ∀ {a b : G}, a s.carrierb s.carriera + b s.carrier
  • zero_mem' : 0 s.carrier
  • neg_mem' : ∀ {x : G}, x s.carrier-x s.carrier
  • isOpen' : IsOpen s.carrier

The type of open subgroups of a topological additive group.

Instances For
    structure OpenSubgroup (G : Type u_1) [Group G] [TopologicalSpace G] extends Subgroup :
    Type u_1
    • carrier : Set G
    • mul_mem' : ∀ {a b : G}, a s.carrierb s.carriera * b s.carrier
    • one_mem' : 1 s.carrier
    • inv_mem' : ∀ {x : G}, x s.carrierx⁻¹ s.carrier
    • isOpen' : IsOpen s.carrier

    The type of open subgroups of a topological group.

    Instances For
      theorem OpenAddSubgroup.toAddSubgroup_injective {G : Type u_1} [AddGroup G] [TopologicalSpace G] :
      Function.Injective OpenAddSubgroup.toAddSubgroup
      abbrev OpenAddSubgroup.toAddSubgroup_injective.match_1 {G : Type u_1} [AddGroup G] [TopologicalSpace G] (motive : (x x_1 : OpenAddSubgroup G) → x = x_1Prop) :
      (x x_1 : OpenAddSubgroup G) → (x_2 : x = x_1) → ((toSubgroup : AddSubgroup G) → (isOpen' isOpen'_1 : IsOpen toSubgroup.carrier) → motive { toAddSubgroup := toSubgroup, isOpen' := isOpen' } { toAddSubgroup := toSubgroup, isOpen' := isOpen'_1 } (_ : { toAddSubgroup := toSubgroup, isOpen' := (_ : IsOpen toSubgroup.carrier) } = { toAddSubgroup := toSubgroup, isOpen' := (_ : IsOpen toSubgroup.carrier) })) → motive x x_1 x_2
      Instances For
        theorem OpenSubgroup.toSubgroup_injective {G : Type u_1} [Group G] [TopologicalSpace G] :
        Function.Injective OpenSubgroup.toSubgroup
        theorem OpenAddSubgroup.instSetLikeOpenAddSubgroup.proof_1 {G : Type u_1} [AddGroup G] [TopologicalSpace G] :
        ∀ (x x_1 : OpenAddSubgroup G), (fun U => U) x = (fun U => U) x_1x = x_1

        Coercion from OpenAddSubgroup G to Opens G.

        Instances For

          Coercion from OpenSubgroup G to Opens G.

          Instances For
            @[simp]
            theorem OpenAddSubgroup.coe_toOpens {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U : OpenAddSubgroup G} :
            U = U
            @[simp]
            theorem OpenSubgroup.coe_toOpens {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.coe_toSubgroup {G : Type u_1} [Group G] [TopologicalSpace G] {U : OpenSubgroup G} :
            U = 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_toOpens {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
            @[simp]
            theorem OpenSubgroup.mem_toSubgroup {G : Type u_1} [Group G] [TopologicalSpace G] {U : OpenSubgroup G} {g : G} :
            g U g U
            theorem OpenAddSubgroup.ext {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U : OpenAddSubgroup G} {V : OpenAddSubgroup G} (h : ∀ (x : G), x U x V) :
            U = V
            theorem OpenSubgroup.ext {G : Type u_1} [Group G] [TopologicalSpace G] {U : OpenSubgroup G} {V : OpenSubgroup G} (h : ∀ (x : G), x U x V) :
            U = V
            @[simp]
            theorem OpenAddSubgroup.mem_top {G : Type u_1} [AddGroup G] [TopologicalSpace G] (x : G) :
            @[simp]
            theorem OpenSubgroup.mem_top {G : Type u_1} [Group G] [TopologicalSpace G] (x : G) :
            @[simp]
            theorem OpenAddSubgroup.coe_top {G : Type u_1} [AddGroup G] [TopologicalSpace G] :
            = Set.univ
            @[simp]
            theorem OpenSubgroup.coe_top {G : Type u_1} [Group G] [TopologicalSpace G] :
            = Set.univ
            @[simp]
            @[simp]

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

            Instances For
              theorem OpenAddSubgroup.sum.proof_1 {G : Type u_1} [AddGroup G] [TopologicalSpace G] {H : Type u_2} [AddGroup H] [TopologicalSpace H] (U : OpenAddSubgroup G) (V : OpenAddSubgroup H) :
              IsOpen (U ×ˢ V.toAddSubmonoid)
              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) :

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

              Instances For
                @[simp]
                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) :
                ↑(OpenAddSubgroup.sum U V) = U ×ˢ V
                @[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) :
                ↑(OpenSubgroup.prod U V) = U ×ˢ V
                @[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) :
                ↑(OpenSubgroup.prod U V) = Subgroup.prod U V
                @[simp]
                theorem OpenAddSubgroup.coe_inf {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U : OpenAddSubgroup G} {V : OpenAddSubgroup G} :
                ↑(U V) = U V
                @[simp]
                theorem OpenSubgroup.coe_inf {G : Type u_1} [Group G] [TopologicalSpace G] {U : OpenSubgroup G} {V : OpenSubgroup G} :
                ↑(U V) = U V
                @[simp]
                theorem OpenAddSubgroup.toAddSubgroup_inf {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U : OpenAddSubgroup G} {V : OpenAddSubgroup G} :
                ↑(U V) = U V
                @[simp]
                theorem OpenSubgroup.toSubgroup_inf {G : Type u_1} [Group G] [TopologicalSpace G] {U : OpenSubgroup G} {V : OpenSubgroup G} :
                ↑(U V) = U V
                @[simp]
                theorem OpenAddSubgroup.toOpens_inf {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U : OpenAddSubgroup G} {V : OpenAddSubgroup G} :
                ↑(U V) = U V
                @[simp]
                theorem OpenSubgroup.toOpens_inf {G : Type u_1} [Group G] [TopologicalSpace G] {U : OpenSubgroup G} {V : OpenSubgroup G} :
                ↑(U V) = U V
                @[simp]
                theorem OpenAddSubgroup.mem_inf {G : Type u_1} [AddGroup G] [TopologicalSpace G] {U : OpenAddSubgroup G} {V : OpenAddSubgroup G} {x : G} :
                x U V x U x V
                @[simp]
                theorem OpenSubgroup.mem_inf {G : Type u_1} [Group G] [TopologicalSpace G] {U : OpenSubgroup G} {V : OpenSubgroup G} {x : G} :
                x U V x U x V
                @[simp]
                @[simp]
                theorem OpenSubgroup.toSubgroup_le {G : Type u_1} [Group G] [TopologicalSpace G] {U : OpenSubgroup G} {V : OpenSubgroup G} :
                U V U V
                theorem OpenAddSubgroup.comap.proof_1 {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) :
                IsOpen (f ⁻¹' H)
                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) :

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

                Instances For
                  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) :

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

                  Instances For
                    @[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) :
                    ↑(OpenAddSubgroup.comap f hf H) = f ⁻¹' H
                    @[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) :
                    ↑(OpenSubgroup.comap f hf H) = f ⁻¹' 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) :
                    @[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) :
                    @[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 OpenAddSubgroup.comap f hf H f x 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 OpenSubgroup.comap f hf H f x H
                    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) :
                    OpenAddSubgroup.comap f₁ hf₁ (OpenAddSubgroup.comap f₂ hf₂ K) = OpenAddSubgroup.comap (AddMonoidHom.comp f₂ f₁) (_ : Continuous (f₂ fun x => f₁ x)) K
                    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) :
                    OpenSubgroup.comap f₁ hf₁ (OpenSubgroup.comap f₂ hf₂ K) = OpenSubgroup.comap (MonoidHom.comp f₂ f₁) (_ : Continuous (f₂ fun x => f₁ x)) K
                    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_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_mono {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] {H₁ : AddSubgroup G} {H₂ : AddSubgroup G} (h : H₁ H₂) (h₁ : IsOpen H₁) :
                    IsOpen H₂
                    theorem Subgroup.isOpen_mono {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousMul G] {H₁ : Subgroup G} {H₂ : Subgroup 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

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

                    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.

                    @[simp]
                    theorem OpenAddSubgroup.toAddSubgroup_sup {G : Type u_1} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] (U : OpenAddSubgroup G) (V : OpenAddSubgroup G) :
                    ↑(U V) = U V
                    @[simp]
                    theorem OpenSubgroup.toSubgroup_sup {G : Type u_1} [Group G] [TopologicalSpace G] [ContinuousMul G] (U : OpenSubgroup G) (V : OpenSubgroup G) :
                    ↑(U V) = U V
                    theorem Submodule.isOpen_mono {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [TopologicalSpace M] [TopologicalAddGroup M] [Module R M] {U : Submodule R M} {P : Submodule R M} (h : U P) (hU : IsOpen U) :
                    IsOpen P
                    theorem Ideal.isOpen_of_open_subideal {R : Type u_1} [CommRing R] [TopologicalSpace R] [TopologicalRing R] {U : Ideal R} {I : Ideal R} (h : U I) (hU : IsOpen U) :
                    IsOpen I