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basic.lean
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basic.lean
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/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import group_theory.group_action.conj_act
import group_theory.group_action.quotient
import group_theory.quotient_group
import order.filter.pointwise
import topology.algebra.monoid
import topology.algebra.constructions
/-!
# Topological groups
This file defines the following typeclasses:
* `topological_group`, `topological_add_group`: multiplicative and additive topological groups,
i.e., groups with continuous `(*)` and `(⁻¹)` / `(+)` and `(-)`;
* `has_continuous_sub G` means that `G` has a continuous subtraction operation.
There is an instance deducing `has_continuous_sub` from `topological_group` but we use a separate
typeclass because, e.g., `ℕ` and `ℝ≥0` have continuous subtraction but are not additive groups.
We also define `homeomorph` versions of several `equiv`s: `homeomorph.mul_left`,
`homeomorph.mul_right`, `homeomorph.inv`, and prove a few facts about neighbourhood filters in
groups.
## Tags
topological space, group, topological group
-/
open classical set filter topological_space function
open_locale classical topological_space filter pointwise
universes u v w x
variables {α : Type u} {β : Type v} {G : Type w} {H : Type x}
section continuous_mul_group
/-!
### Groups with continuous multiplication
In this section we prove a few statements about groups with continuous `(*)`.
-/
variables [topological_space G] [group G] [has_continuous_mul G]
/-- Multiplication from the left in a topological group as a homeomorphism. -/
@[to_additive "Addition from the left in a topological additive group as a homeomorphism."]
protected def homeomorph.mul_left (a : G) : G ≃ₜ G :=
{ continuous_to_fun := continuous_const.mul continuous_id,
continuous_inv_fun := continuous_const.mul continuous_id,
.. equiv.mul_left a }
@[simp, to_additive]
lemma homeomorph.coe_mul_left (a : G) : ⇑(homeomorph.mul_left a) = (*) a := rfl
@[to_additive]
lemma homeomorph.mul_left_symm (a : G) : (homeomorph.mul_left a).symm = homeomorph.mul_left a⁻¹ :=
by { ext, refl }
@[to_additive]
lemma is_open_map_mul_left (a : G) : is_open_map (λ x, a * x) :=
(homeomorph.mul_left a).is_open_map
@[to_additive is_open.left_add_coset]
lemma is_open.left_coset {U : set G} (h : is_open U) (x : G) : is_open (left_coset x U) :=
is_open_map_mul_left x _ h
@[to_additive]
lemma is_closed_map_mul_left (a : G) : is_closed_map (λ x, a * x) :=
(homeomorph.mul_left a).is_closed_map
@[to_additive is_closed.left_add_coset]
lemma is_closed.left_coset {U : set G} (h : is_closed U) (x : G) : is_closed (left_coset x U) :=
is_closed_map_mul_left x _ h
/-- Multiplication from the right in a topological group as a homeomorphism. -/
@[to_additive "Addition from the right in a topological additive group as a homeomorphism."]
protected def homeomorph.mul_right (a : G) :
G ≃ₜ G :=
{ continuous_to_fun := continuous_id.mul continuous_const,
continuous_inv_fun := continuous_id.mul continuous_const,
.. equiv.mul_right a }
@[simp, to_additive]
lemma homeomorph.coe_mul_right (a : G) : ⇑(homeomorph.mul_right a) = λ g, g * a := rfl
@[to_additive]
lemma homeomorph.mul_right_symm (a : G) :
(homeomorph.mul_right a).symm = homeomorph.mul_right a⁻¹ :=
by { ext, refl }
@[to_additive]
lemma is_open_map_mul_right (a : G) : is_open_map (λ x, x * a) :=
(homeomorph.mul_right a).is_open_map
@[to_additive is_open.right_add_coset]
lemma is_open.right_coset {U : set G} (h : is_open U) (x : G) : is_open (right_coset U x) :=
is_open_map_mul_right x _ h
@[to_additive]
lemma is_closed_map_mul_right (a : G) : is_closed_map (λ x, x * a) :=
(homeomorph.mul_right a).is_closed_map
@[to_additive is_closed.right_add_coset]
lemma is_closed.right_coset {U : set G} (h : is_closed U) (x : G) : is_closed (right_coset U x) :=
is_closed_map_mul_right x _ h
@[to_additive]
lemma discrete_topology_of_open_singleton_one (h : is_open ({1} : set G)) : discrete_topology G :=
begin
rw ← singletons_open_iff_discrete,
intro g,
suffices : {g} = (λ (x : G), g⁻¹ * x) ⁻¹' {1},
{ rw this, exact (continuous_mul_left (g⁻¹)).is_open_preimage _ h, },
simp only [mul_one, set.preimage_mul_left_singleton, eq_self_iff_true,
inv_inv, set.singleton_eq_singleton_iff],
end
@[to_additive]
lemma discrete_topology_iff_open_singleton_one : discrete_topology G ↔ is_open ({1} : set G) :=
⟨λ h, forall_open_iff_discrete.mpr h {1}, discrete_topology_of_open_singleton_one⟩
end continuous_mul_group
/-!
### `has_continuous_inv` and `has_continuous_neg`
-/
/-- Basic hypothesis to talk about a topological additive group. A topological additive group
over `M`, for example, is obtained by requiring the instances `add_group M` and
`has_continuous_add M` and `has_continuous_neg M`. -/
class has_continuous_neg (G : Type u) [topological_space G] [has_neg G] : Prop :=
(continuous_neg : continuous (λ a : G, -a))
/-- Basic hypothesis to talk about a topological group. A topological group over `M`, for example,
is obtained by requiring the instances `group M` and `has_continuous_mul M` and
`has_continuous_inv M`. -/
@[to_additive]
class has_continuous_inv (G : Type u) [topological_space G] [has_inv G] : Prop :=
(continuous_inv : continuous (λ a : G, a⁻¹))
export has_continuous_inv (continuous_inv)
export has_continuous_neg (continuous_neg)
section continuous_inv
variables [topological_space G] [has_inv G] [has_continuous_inv G]
@[to_additive]
lemma continuous_on_inv {s : set G} : continuous_on has_inv.inv s :=
continuous_inv.continuous_on
@[to_additive]
lemma continuous_within_at_inv {s : set G} {x : G} : continuous_within_at has_inv.inv s x :=
continuous_inv.continuous_within_at
@[to_additive]
lemma continuous_at_inv {x : G} : continuous_at has_inv.inv x :=
continuous_inv.continuous_at
@[to_additive]
lemma tendsto_inv (a : G) : tendsto has_inv.inv (𝓝 a) (𝓝 (a⁻¹)) :=
continuous_at_inv
/-- If a function converges to a value in a multiplicative topological group, then its inverse
converges to the inverse of this value. For the version in normed fields assuming additionally
that the limit is nonzero, use `tendsto.inv'`. -/
@[to_additive "If a function converges to a value in an additive topological group, then its
negation converges to the negation of this value."]
lemma filter.tendsto.inv {f : α → G} {l : filter α} {y : G} (h : tendsto f l (𝓝 y)) :
tendsto (λ x, (f x)⁻¹) l (𝓝 y⁻¹) :=
(continuous_inv.tendsto y).comp h
variables [topological_space α] {f : α → G} {s : set α} {x : α}
@[continuity, to_additive]
lemma continuous.inv (hf : continuous f) : continuous (λx, (f x)⁻¹) :=
continuous_inv.comp hf
@[to_additive]
lemma continuous_at.inv (hf : continuous_at f x) : continuous_at (λ x, (f x)⁻¹) x :=
continuous_at_inv.comp hf
@[to_additive]
lemma continuous_on.inv (hf : continuous_on f s) : continuous_on (λx, (f x)⁻¹) s :=
continuous_inv.comp_continuous_on hf
@[to_additive]
lemma continuous_within_at.inv (hf : continuous_within_at f s x) :
continuous_within_at (λ x, (f x)⁻¹) s x :=
hf.inv
@[to_additive]
instance [topological_space H] [has_inv H] [has_continuous_inv H] : has_continuous_inv (G × H) :=
⟨continuous_inv.fst'.prod_mk continuous_inv.snd'⟩
variable {ι : Type*}
@[to_additive]
instance pi.has_continuous_inv {C : ι → Type*} [∀ i, topological_space (C i)]
[∀ i, has_inv (C i)] [∀ i, has_continuous_inv (C i)] : has_continuous_inv (Π i, C i) :=
{ continuous_inv := continuous_pi (λ i, (continuous_apply i).inv) }
/-- A version of `pi.has_continuous_inv` for non-dependent functions. It is needed because sometimes
Lean fails to use `pi.has_continuous_inv` for non-dependent functions. -/
@[to_additive "A version of `pi.has_continuous_neg` for non-dependent functions. It is needed
because sometimes Lean fails to use `pi.has_continuous_neg` for non-dependent functions."]
instance pi.has_continuous_inv' : has_continuous_inv (ι → G) :=
pi.has_continuous_inv
@[priority 100, to_additive]
instance has_continuous_inv_of_discrete_topology [topological_space H]
[has_inv H] [discrete_topology H] : has_continuous_inv H :=
⟨continuous_of_discrete_topology⟩
section pointwise_limits
variables (G₁ G₂ : Type*) [topological_space G₂] [t2_space G₂]
@[to_additive] lemma is_closed_set_of_map_inv [has_inv G₁] [has_inv G₂] [has_continuous_inv G₂] :
is_closed {f : G₁ → G₂ | ∀ x, f x⁻¹ = (f x)⁻¹ } :=
begin
simp only [set_of_forall],
refine is_closed_Inter (λ i, is_closed_eq (continuous_apply _) (continuous_apply _).inv),
end
end pointwise_limits
instance [topological_space H] [has_inv H] [has_continuous_inv H] :
has_continuous_neg (additive H) :=
{ continuous_neg := @continuous_inv H _ _ _ }
instance [topological_space H] [has_neg H] [has_continuous_neg H] :
has_continuous_inv (multiplicative H) :=
{ continuous_inv := @continuous_neg H _ _ _ }
end continuous_inv
section continuous_involutive_inv
variables [topological_space G] [has_involutive_inv G] [has_continuous_inv G] {s : set G}
@[to_additive] lemma is_compact.inv (hs : is_compact s) : is_compact s⁻¹ :=
by { rw [← image_inv], exact hs.image continuous_inv }
variables (G)
/-- Inversion in a topological group as a homeomorphism. -/
@[to_additive "Negation in a topological group as a homeomorphism."]
protected def homeomorph.inv (G : Type*) [topological_space G] [has_involutive_inv G]
[has_continuous_inv G] : G ≃ₜ G :=
{ continuous_to_fun := continuous_inv,
continuous_inv_fun := continuous_inv,
.. equiv.inv G }
@[to_additive] lemma is_open_map_inv : is_open_map (has_inv.inv : G → G) :=
(homeomorph.inv _).is_open_map
@[to_additive] lemma is_closed_map_inv : is_closed_map (has_inv.inv : G → G) :=
(homeomorph.inv _).is_closed_map
variables {G}
@[to_additive] lemma is_open.inv (hs : is_open s) : is_open s⁻¹ := hs.preimage continuous_inv
@[to_additive] lemma is_closed.inv (hs : is_closed s) : is_closed s⁻¹ := hs.preimage continuous_inv
@[to_additive] lemma inv_closure : ∀ s : set G, (closure s)⁻¹ = closure s⁻¹ :=
(homeomorph.inv G).preimage_closure
end continuous_involutive_inv
section lattice_ops
variables {ι' : Sort*} [has_inv G]
@[to_additive] lemma has_continuous_inv_Inf {ts : set (topological_space G)}
(h : Π t ∈ ts, @has_continuous_inv G t _) :
@has_continuous_inv G (Inf ts) _ :=
{ continuous_inv := continuous_Inf_rng.2 (λ t ht, continuous_Inf_dom ht
(@has_continuous_inv.continuous_inv G t _ (h t ht))) }
@[to_additive] lemma has_continuous_inv_infi {ts' : ι' → topological_space G}
(h' : Π i, @has_continuous_inv G (ts' i) _) :
@has_continuous_inv G (⨅ i, ts' i) _ :=
by {rw ← Inf_range, exact has_continuous_inv_Inf (set.forall_range_iff.mpr h')}
@[to_additive] lemma has_continuous_inv_inf {t₁ t₂ : topological_space G}
(h₁ : @has_continuous_inv G t₁ _) (h₂ : @has_continuous_inv G t₂ _) :
@has_continuous_inv G (t₁ ⊓ t₂) _ :=
by { rw inf_eq_infi, refine has_continuous_inv_infi (λ b, _), cases b; assumption }
end lattice_ops
@[to_additive] lemma inducing.has_continuous_inv {G H : Type*} [has_inv G] [has_inv H]
[topological_space G] [topological_space H] [has_continuous_inv H] {f : G → H} (hf : inducing f)
(hf_inv : ∀ x, f x⁻¹ = (f x)⁻¹) : has_continuous_inv G :=
⟨hf.continuous_iff.2 $ by simpa only [(∘), hf_inv] using hf.continuous.inv⟩
section topological_group
/-!
### Topological groups
A topological group is a group in which the multiplication and inversion operations are
continuous. Topological additive groups are defined in the same way. Equivalently, we can require
that the division operation `λ x y, x * y⁻¹` (resp., subtraction) is continuous.
-/
/-- A topological (additive) group is a group in which the addition and negation operations are
continuous. -/
class topological_add_group (G : Type u) [topological_space G] [add_group G]
extends has_continuous_add G, has_continuous_neg G : Prop
/-- A topological group is a group in which the multiplication and inversion operations are
continuous.
When you declare an instance that does not already have a `uniform_space` instance,
you should also provide an instance of `uniform_space` and `uniform_group` using
`topological_group.to_uniform_space` and `topological_comm_group_is_uniform`. -/
@[to_additive]
class topological_group (G : Type*) [topological_space G] [group G]
extends has_continuous_mul G, has_continuous_inv G : Prop
section conj
instance conj_act.units_has_continuous_const_smul {M} [monoid M] [topological_space M]
[has_continuous_mul M] :
has_continuous_const_smul (conj_act Mˣ) M :=
⟨λ m, (continuous_const.mul continuous_id).mul continuous_const⟩
/-- we slightly weaken the type class assumptions here so that it will also apply to `ennreal`, but
we nevertheless leave it in the `topological_group` namespace. -/
variables [topological_space G] [has_inv G] [has_mul G] [has_continuous_mul G]
/-- Conjugation is jointly continuous on `G × G` when both `mul` and `inv` are continuous. -/
@[to_additive "Conjugation is jointly continuous on `G × G` when both `mul` and `inv` are
continuous."]
lemma topological_group.continuous_conj_prod [has_continuous_inv G] :
continuous (λ g : G × G, g.fst * g.snd * g.fst⁻¹) :=
continuous_mul.mul (continuous_inv.comp continuous_fst)
/-- Conjugation by a fixed element is continuous when `mul` is continuous. -/
@[to_additive "Conjugation by a fixed element is continuous when `add` is continuous."]
lemma topological_group.continuous_conj (g : G) : continuous (λ (h : G), g * h * g⁻¹) :=
(continuous_mul_right g⁻¹).comp (continuous_mul_left g)
/-- Conjugation acting on fixed element of the group is continuous when both `mul` and
`inv` are continuous. -/
@[to_additive "Conjugation acting on fixed element of the additive group is continuous when both
`add` and `neg` are continuous."]
lemma topological_group.continuous_conj' [has_continuous_inv G]
(h : G) : continuous (λ (g : G), g * h * g⁻¹) :=
(continuous_mul_right h).mul continuous_inv
end conj
variables [topological_space G] [group G] [topological_group G]
[topological_space α] {f : α → G} {s : set α} {x : α}
section zpow
@[continuity, to_additive]
lemma continuous_zpow : ∀ z : ℤ, continuous (λ a : G, a ^ z)
| (int.of_nat n) := by simpa using continuous_pow n
| -[1+n] := by simpa using (continuous_pow (n + 1)).inv
instance add_group.has_continuous_const_smul_int {A} [add_group A] [topological_space A]
[topological_add_group A] : has_continuous_const_smul ℤ A := ⟨continuous_zsmul⟩
instance add_group.has_continuous_smul_int {A} [add_group A] [topological_space A]
[topological_add_group A] : has_continuous_smul ℤ A :=
⟨continuous_uncurry_of_discrete_topology continuous_zsmul⟩
@[continuity, to_additive]
lemma continuous.zpow {f : α → G} (h : continuous f) (z : ℤ) :
continuous (λ b, (f b) ^ z) :=
(continuous_zpow z).comp h
@[to_additive]
lemma continuous_on_zpow {s : set G} (z : ℤ) : continuous_on (λ x, x ^ z) s :=
(continuous_zpow z).continuous_on
@[to_additive]
lemma continuous_at_zpow (x : G) (z : ℤ) : continuous_at (λ x, x ^ z) x :=
(continuous_zpow z).continuous_at
@[to_additive]
lemma filter.tendsto.zpow {α} {l : filter α} {f : α → G} {x : G} (hf : tendsto f l (𝓝 x)) (z : ℤ) :
tendsto (λ x, f x ^ z) l (𝓝 (x ^ z)) :=
(continuous_at_zpow _ _).tendsto.comp hf
@[to_additive]
lemma continuous_within_at.zpow {f : α → G} {x : α} {s : set α} (hf : continuous_within_at f s x)
(z : ℤ) : continuous_within_at (λ x, f x ^ z) s x :=
hf.zpow z
@[to_additive]
lemma continuous_at.zpow {f : α → G} {x : α} (hf : continuous_at f x) (z : ℤ) :
continuous_at (λ x, f x ^ z) x :=
hf.zpow z
@[to_additive continuous_on.zsmul]
lemma continuous_on.zpow {f : α → G} {s : set α} (hf : continuous_on f s) (z : ℤ) :
continuous_on (λ x, f x ^ z) s :=
λ x hx, (hf x hx).zpow z
end zpow
section ordered_comm_group
variables [topological_space H] [ordered_comm_group H] [topological_group H]
@[to_additive] lemma tendsto_inv_nhds_within_Ioi {a : H} :
tendsto has_inv.inv (𝓝[>] a) (𝓝[<] (a⁻¹)) :=
(continuous_inv.tendsto a).inf $ by simp [tendsto_principal_principal]
@[to_additive] lemma tendsto_inv_nhds_within_Iio {a : H} :
tendsto has_inv.inv (𝓝[<] a) (𝓝[>] (a⁻¹)) :=
(continuous_inv.tendsto a).inf $ by simp [tendsto_principal_principal]
@[to_additive] lemma tendsto_inv_nhds_within_Ioi_inv {a : H} :
tendsto has_inv.inv (𝓝[>] (a⁻¹)) (𝓝[<] a) :=
by simpa only [inv_inv] using @tendsto_inv_nhds_within_Ioi _ _ _ _ (a⁻¹)
@[to_additive] lemma tendsto_inv_nhds_within_Iio_inv {a : H} :
tendsto has_inv.inv (𝓝[<] (a⁻¹)) (𝓝[>] a) :=
by simpa only [inv_inv] using @tendsto_inv_nhds_within_Iio _ _ _ _ (a⁻¹)
@[to_additive] lemma tendsto_inv_nhds_within_Ici {a : H} :
tendsto has_inv.inv (𝓝[≥] a) (𝓝[≤] (a⁻¹)) :=
(continuous_inv.tendsto a).inf $ by simp [tendsto_principal_principal]
@[to_additive] lemma tendsto_inv_nhds_within_Iic {a : H} :
tendsto has_inv.inv (𝓝[≤] a) (𝓝[≥] (a⁻¹)) :=
(continuous_inv.tendsto a).inf $ by simp [tendsto_principal_principal]
@[to_additive] lemma tendsto_inv_nhds_within_Ici_inv {a : H} :
tendsto has_inv.inv (𝓝[≥] (a⁻¹)) (𝓝[≤] a) :=
by simpa only [inv_inv] using @tendsto_inv_nhds_within_Ici _ _ _ _ (a⁻¹)
@[to_additive] lemma tendsto_inv_nhds_within_Iic_inv {a : H} :
tendsto has_inv.inv (𝓝[≤] (a⁻¹)) (𝓝[≥] a) :=
by simpa only [inv_inv] using @tendsto_inv_nhds_within_Iic _ _ _ _ (a⁻¹)
end ordered_comm_group
@[instance, to_additive]
instance [topological_space H] [group H] [topological_group H] :
topological_group (G × H) :=
{ continuous_inv := continuous_inv.prod_map continuous_inv }
@[to_additive]
instance pi.topological_group {C : β → Type*} [∀ b, topological_space (C b)]
[∀ b, group (C b)] [∀ b, topological_group (C b)] : topological_group (Π b, C b) :=
{ continuous_inv := continuous_pi (λ i, (continuous_apply i).inv) }
open mul_opposite
@[to_additive]
instance [group α] [has_continuous_inv α] : has_continuous_inv αᵐᵒᵖ :=
op_homeomorph.symm.inducing.has_continuous_inv unop_inv
/-- If multiplication is continuous in `α`, then it also is in `αᵐᵒᵖ`. -/
@[to_additive "If addition is continuous in `α`, then it also is in `αᵃᵒᵖ`."]
instance [group α] [topological_group α] :
topological_group αᵐᵒᵖ := { }
variable (G)
@[to_additive]
lemma nhds_one_symm : comap has_inv.inv (𝓝 (1 : G)) = 𝓝 (1 : G) :=
((homeomorph.inv G).comap_nhds_eq _).trans (congr_arg nhds inv_one)
/-- The map `(x, y) ↦ (x, xy)` as a homeomorphism. This is a shear mapping. -/
@[to_additive "The map `(x, y) ↦ (x, x + y)` as a homeomorphism.
This is a shear mapping."]
protected def homeomorph.shear_mul_right : G × G ≃ₜ G × G :=
{ continuous_to_fun := continuous_fst.prod_mk continuous_mul,
continuous_inv_fun := continuous_fst.prod_mk $ continuous_fst.inv.mul continuous_snd,
.. equiv.prod_shear (equiv.refl _) equiv.mul_left }
@[simp, to_additive]
lemma homeomorph.shear_mul_right_coe :
⇑(homeomorph.shear_mul_right G) = λ z : G × G, (z.1, z.1 * z.2) :=
rfl
@[simp, to_additive]
lemma homeomorph.shear_mul_right_symm_coe :
⇑(homeomorph.shear_mul_right G).symm = λ z : G × G, (z.1, z.1⁻¹ * z.2) :=
rfl
variables {G}
@[to_additive] protected lemma inducing.topological_group {F : Type*} [group H]
[topological_space H] [monoid_hom_class F H G] (f : F) (hf : inducing f) :
topological_group H :=
{ to_has_continuous_mul := hf.has_continuous_mul _,
to_has_continuous_inv := hf.has_continuous_inv (map_inv f) }
@[to_additive] protected lemma topological_group_induced {F : Type*} [group H]
[monoid_hom_class F H G] (f : F) :
@topological_group H (induced f ‹_›) _ :=
by { letI := induced f ‹_›, exact inducing.topological_group f ⟨rfl⟩ }
namespace subgroup
@[to_additive] instance (S : subgroup G) : topological_group S :=
inducing.topological_group S.subtype inducing_coe
end subgroup
/-- The (topological-space) closure of a subgroup of a space `M` with `has_continuous_mul` is
itself a subgroup. -/
@[to_additive "The (topological-space) closure of an additive subgroup of a space `M` with
`has_continuous_add` is itself an additive subgroup."]
def subgroup.topological_closure (s : subgroup G) : subgroup G :=
{ carrier := closure (s : set G),
inv_mem' := λ g m, by simpa [←set.mem_inv, inv_closure] using m,
..s.to_submonoid.topological_closure }
@[simp, to_additive] lemma subgroup.topological_closure_coe {s : subgroup G} :
(s.topological_closure : set G) = closure s :=
rfl
@[to_additive] lemma subgroup.subgroup_topological_closure (s : subgroup G) :
s ≤ s.topological_closure :=
subset_closure
@[to_additive] lemma subgroup.is_closed_topological_closure (s : subgroup G) :
is_closed (s.topological_closure : set G) :=
by convert is_closed_closure
@[to_additive] lemma subgroup.topological_closure_minimal
(s : subgroup G) {t : subgroup G} (h : s ≤ t) (ht : is_closed (t : set G)) :
s.topological_closure ≤ t :=
closure_minimal h ht
@[to_additive] lemma dense_range.topological_closure_map_subgroup [group H] [topological_space H]
[topological_group H] {f : G →* H} (hf : continuous f) (hf' : dense_range f) {s : subgroup G}
(hs : s.topological_closure = ⊤) :
(s.map f).topological_closure = ⊤ :=
begin
rw set_like.ext'_iff at hs ⊢,
simp only [subgroup.topological_closure_coe, subgroup.coe_top, ← dense_iff_closure_eq] at hs ⊢,
exact hf'.dense_image hf hs
end
/-- The topological closure of a normal subgroup is normal.-/
@[to_additive "The topological closure of a normal additive subgroup is normal."]
lemma subgroup.is_normal_topological_closure {G : Type*} [topological_space G] [group G]
[topological_group G] (N : subgroup G) [N.normal] :
(subgroup.topological_closure N).normal :=
{ conj_mem := λ n hn g,
begin
apply map_mem_closure (topological_group.continuous_conj g) hn,
exact λ m hm, subgroup.normal.conj_mem infer_instance m hm g
end }
@[to_additive] lemma mul_mem_connected_component_one {G : Type*} [topological_space G]
[mul_one_class G] [has_continuous_mul G] {g h : G} (hg : g ∈ connected_component (1 : G))
(hh : h ∈ connected_component (1 : G)) : g * h ∈ connected_component (1 : G) :=
begin
rw connected_component_eq hg,
have hmul: g ∈ connected_component (g*h),
{ apply continuous.image_connected_component_subset (continuous_mul_left g),
rw ← connected_component_eq hh,
exact ⟨(1 : G), mem_connected_component, by simp only [mul_one]⟩ },
simpa [← connected_component_eq hmul] using (mem_connected_component)
end
@[to_additive] lemma inv_mem_connected_component_one {G : Type*} [topological_space G] [group G]
[topological_group G] {g : G} (hg : g ∈ connected_component (1 : G)) :
g⁻¹ ∈ connected_component (1 : G) :=
begin
rw ← inv_one,
exact continuous.image_connected_component_subset continuous_inv _
((set.mem_image _ _ _).mp ⟨g, hg, rfl⟩)
end
/-- The connected component of 1 is a subgroup of `G`. -/
@[to_additive "The connected component of 0 is a subgroup of `G`."]
def subgroup.connected_component_of_one (G : Type*) [topological_space G] [group G]
[topological_group G] : subgroup G :=
{ carrier := connected_component (1 : G),
one_mem' := mem_connected_component,
mul_mem' := λ g h hg hh, mul_mem_connected_component_one hg hh,
inv_mem' := λ g hg, inv_mem_connected_component_one hg }
/-- If a subgroup of a topological group is commutative, then so is its topological closure. -/
@[to_additive "If a subgroup of an additive topological group is commutative, then so is its
topological closure."]
def subgroup.comm_group_topological_closure [t2_space G] (s : subgroup G)
(hs : ∀ (x y : s), x * y = y * x) : comm_group s.topological_closure :=
{ ..s.topological_closure.to_group,
..s.to_submonoid.comm_monoid_topological_closure hs }
@[to_additive exists_nhds_half_neg]
lemma exists_nhds_split_inv {s : set G} (hs : s ∈ 𝓝 (1 : G)) :
∃ V ∈ 𝓝 (1 : G), ∀ (v ∈ V) (w ∈ V), v / w ∈ s :=
have ((λp : G × G, p.1 * p.2⁻¹) ⁻¹' s) ∈ 𝓝 ((1, 1) : G × G),
from continuous_at_fst.mul continuous_at_snd.inv (by simpa),
by simpa only [div_eq_mul_inv, nhds_prod_eq, mem_prod_self_iff, prod_subset_iff, mem_preimage]
using this
@[to_additive]
lemma nhds_translation_mul_inv (x : G) : comap (λ y : G, y * x⁻¹) (𝓝 1) = 𝓝 x :=
((homeomorph.mul_right x⁻¹).comap_nhds_eq 1).trans $ show 𝓝 (1 * x⁻¹⁻¹) = 𝓝 x, by simp
@[simp, to_additive] lemma map_mul_left_nhds (x y : G) : map ((*) x) (𝓝 y) = 𝓝 (x * y) :=
(homeomorph.mul_left x).map_nhds_eq y
@[to_additive] lemma map_mul_left_nhds_one (x : G) : map ((*) x) (𝓝 1) = 𝓝 x := by simp
/-- A monoid homomorphism (a bundled morphism of a type that implements `monoid_hom_class`) from a
topological group to a topological monoid is continuous provided that it is continuous at one. See
also `uniform_continuous_of_continuous_at_one`. -/
@[to_additive "An additive monoid homomorphism (a bundled morphism of a type that implements
`add_monoid_hom_class`) from an additive topological group to an additive topological monoid is
continuous provided that it is continuous at zero. See also
`uniform_continuous_of_continuous_at_zero`."]
lemma continuous_of_continuous_at_one {M hom : Type*} [mul_one_class M] [topological_space M]
[has_continuous_mul M] [monoid_hom_class hom G M] (f : hom) (hf : continuous_at f 1) :
continuous f :=
continuous_iff_continuous_at.2 $ λ x,
by simpa only [continuous_at, ← map_mul_left_nhds_one x, tendsto_map'_iff, (∘),
map_mul, map_one, mul_one] using hf.tendsto.const_mul (f x)
@[to_additive]
lemma topological_group.ext {G : Type*} [group G] {t t' : topological_space G}
(tg : @topological_group G t _) (tg' : @topological_group G t' _)
(h : @nhds G t 1 = @nhds G t' 1) : t = t' :=
eq_of_nhds_eq_nhds $ λ x, by
rw [← @nhds_translation_mul_inv G t _ _ x , ← @nhds_translation_mul_inv G t' _ _ x , ← h]
@[to_additive]
lemma topological_group.ext_iff {G : Type*} [group G] {t t' : topological_space G}
(tg : @topological_group G t _) (tg' : @topological_group G t' _) :
t = t' ↔ @nhds G t 1 = @nhds G t' 1 :=
⟨λ h, h ▸ rfl, tg.ext tg'⟩
@[to_additive]
lemma topological_group.of_nhds_aux {G : Type*} [group G] [topological_space G]
(hinv : tendsto (λ (x : G), x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ (x₀ : G), 𝓝 x₀ = map (λ (x : G), x₀ * x) (𝓝 1))
(hconj : ∀ (x₀ : G), map (λ (x : G), x₀ * x * x₀⁻¹) (𝓝 1) ≤ 𝓝 1) : continuous (λ x : G, x⁻¹) :=
begin
rw continuous_iff_continuous_at,
rintros x₀,
have key : (λ x, (x₀*x)⁻¹) = (λ x, x₀⁻¹*x) ∘ (λ x, x₀*x*x₀⁻¹) ∘ (λ x, x⁻¹),
by {ext ; simp[mul_assoc] },
calc map (λ x, x⁻¹) (𝓝 x₀)
= map (λ x, x⁻¹) (map (λ x, x₀*x) $ 𝓝 1) : by rw hleft
... = map (λ x, (x₀*x)⁻¹) (𝓝 1) : by rw filter.map_map
... = map (((λ x, x₀⁻¹*x) ∘ (λ x, x₀*x*x₀⁻¹)) ∘ (λ x, x⁻¹)) (𝓝 1) : by rw key
... = map ((λ x, x₀⁻¹*x) ∘ (λ x, x₀*x*x₀⁻¹)) _ : by rw ← filter.map_map
... ≤ map ((λ x, x₀⁻¹ * x) ∘ λ x, x₀ * x * x₀⁻¹) (𝓝 1) : map_mono hinv
... = map (λ x, x₀⁻¹ * x) (map (λ x, x₀ * x * x₀⁻¹) (𝓝 1)) : filter.map_map
... ≤ map (λ x, x₀⁻¹ * x) (𝓝 1) : map_mono (hconj x₀)
... = 𝓝 x₀⁻¹ : (hleft _).symm
end
@[to_additive]
lemma topological_group.of_nhds_one' {G : Type u} [group G] [topological_space G]
(hmul : tendsto (uncurry ((*) : G → G → G)) ((𝓝 1) ×ᶠ 𝓝 1) (𝓝 1))
(hinv : tendsto (λ x : G, x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (λ x, x₀*x) (𝓝 1))
(hright : ∀ x₀ : G, 𝓝 x₀ = map (λ x, x*x₀) (𝓝 1)) : topological_group G :=
begin
refine { continuous_mul := (has_continuous_mul.of_nhds_one hmul hleft hright).continuous_mul,
continuous_inv := topological_group.of_nhds_aux hinv hleft _ },
intros x₀,
suffices : map (λ (x : G), x₀ * x * x₀⁻¹) (𝓝 1) = 𝓝 1, by simp [this, le_refl],
rw [show (λ x, x₀ * x * x₀⁻¹) = (λ x, x₀ * x) ∘ λ x, x*x₀⁻¹, by {ext, simp [mul_assoc] },
← filter.map_map, ← hright, hleft x₀⁻¹, filter.map_map],
convert map_id,
ext,
simp
end
@[to_additive]
lemma topological_group.of_nhds_one {G : Type u} [group G] [topological_space G]
(hmul : tendsto (uncurry ((*) : G → G → G)) ((𝓝 1) ×ᶠ 𝓝 1) (𝓝 1))
(hinv : tendsto (λ x : G, x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (λ x, x₀*x) (𝓝 1))
(hconj : ∀ x₀ : G, tendsto (λ x, x₀*x*x₀⁻¹) (𝓝 1) (𝓝 1)) : topological_group G :=
{ continuous_mul := begin
rw continuous_iff_continuous_at,
rintros ⟨x₀, y₀⟩,
have key : (λ (p : G × G), x₀ * p.1 * (y₀ * p.2)) =
((λ x, x₀*y₀*x) ∘ (uncurry (*)) ∘ (prod.map (λ x, y₀⁻¹*x*y₀) id)),
by { ext, simp [uncurry, prod.map, mul_assoc] },
specialize hconj y₀⁻¹, rw inv_inv at hconj,
calc map (λ (p : G × G), p.1 * p.2) (𝓝 (x₀, y₀))
= map (λ (p : G × G), p.1 * p.2) ((𝓝 x₀) ×ᶠ 𝓝 y₀)
: by rw nhds_prod_eq
... = map (λ (p : G × G), x₀ * p.1 * (y₀ * p.2)) ((𝓝 1) ×ᶠ (𝓝 1))
: by rw [hleft x₀, hleft y₀, prod_map_map_eq, filter.map_map]
... = map (((λ x, x₀*y₀*x) ∘ (uncurry (*))) ∘ (prod.map (λ x, y₀⁻¹*x*y₀) id))((𝓝 1) ×ᶠ (𝓝 1))
: by rw key
... = map ((λ x, x₀*y₀*x) ∘ (uncurry (*))) ((map (λ x, y₀⁻¹*x*y₀) $ 𝓝 1) ×ᶠ (𝓝 1))
: by rw [← filter.map_map, ← prod_map_map_eq', map_id]
... ≤ map ((λ x, x₀*y₀*x) ∘ (uncurry (*))) ((𝓝 1) ×ᶠ (𝓝 1))
: map_mono (filter.prod_mono hconj $ le_rfl)
... = map (λ x, x₀*y₀*x) (map (uncurry (*)) ((𝓝 1) ×ᶠ (𝓝 1))) : by rw filter.map_map
... ≤ map (λ x, x₀*y₀*x) (𝓝 1) : map_mono hmul
... = 𝓝 (x₀*y₀) : (hleft _).symm
end,
continuous_inv := topological_group.of_nhds_aux hinv hleft hconj}
@[to_additive]
lemma topological_group.of_comm_of_nhds_one {G : Type u} [comm_group G] [topological_space G]
(hmul : tendsto (uncurry ((*) : G → G → G)) ((𝓝 1) ×ᶠ 𝓝 1) (𝓝 1))
(hinv : tendsto (λ x : G, x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (λ x, x₀*x) (𝓝 1)) : topological_group G :=
topological_group.of_nhds_one hmul hinv hleft (by simpa using tendsto_id)
end topological_group
section quotient_topological_group
variables [topological_space G] [group G] [topological_group G] (N : subgroup G) (n : N.normal)
@[to_additive]
instance quotient_group.quotient.topological_space {G : Type*} [group G] [topological_space G]
(N : subgroup G) : topological_space (G ⧸ N) :=
quotient.topological_space
open quotient_group
@[to_additive]
lemma quotient_group.is_open_map_coe : is_open_map (coe : G → G ⧸ N) :=
begin
intros s s_op,
change is_open ((coe : G → G ⧸ N) ⁻¹' (coe '' s)),
rw quotient_group.preimage_image_coe N s,
exact is_open_Union (λ n, (continuous_mul_right _).is_open_preimage s s_op)
end
@[to_additive]
instance topological_group_quotient [N.normal] : topological_group (G ⧸ N) :=
{ continuous_mul := begin
have cont : continuous ((coe : G → G ⧸ N) ∘ (λ (p : G × G), p.fst * p.snd)) :=
continuous_quot_mk.comp continuous_mul,
have quot : quotient_map (λ p : G × G, ((p.1 : G ⧸ N), (p.2 : G ⧸ N))),
{ apply is_open_map.to_quotient_map,
{ exact (quotient_group.is_open_map_coe N).prod (quotient_group.is_open_map_coe N) },
{ exact continuous_quot_mk.prod_map continuous_quot_mk },
{ exact (surjective_quot_mk _).prod_map (surjective_quot_mk _) } },
exact (quotient_map.continuous_iff quot).2 cont,
end,
continuous_inv := by convert (@continuous_inv G _ _ _).quotient_map' _ }
/-- Neighborhoods in the quotient are precisely the map of neighborhoods in the prequotient. -/
@[to_additive "Neighborhoods in the quotient are precisely the map of neighborhoods in
the prequotient."]
lemma quotient_group.nhds_eq (x : G) : 𝓝 (x : G ⧸ N) = map coe (𝓝 x) :=
le_antisymm ((quotient_group.is_open_map_coe N).nhds_le x) continuous_quot_mk.continuous_at
variables (G) [first_countable_topology G]
/-- Any first countable topological group has an antitone neighborhood basis `u : ℕ → set G` for
which `(u (n + 1)) ^ 2 ⊆ u n`. The existence of such a neighborhood basis is a key tool for
`quotient_group.complete_space` -/
@[to_additive "Any first countable topological additive group has an antitone neighborhood basis
`u : ℕ → set G` for which `u (n + 1) + u (n + 1) ⊆ u n`. The existence of such a neighborhood basis
is a key tool for `quotient_add_group.complete_space`"]
lemma topological_group.exists_antitone_basis_nhds_one :
∃ (u : ℕ → set G), (𝓝 1).has_antitone_basis u ∧ (∀ n, u (n + 1) * u (n + 1) ⊆ u n) :=
begin
rcases (𝓝 (1 : G)).exists_antitone_basis with ⟨u, hu, u_anti⟩,
have := ((hu.prod_nhds hu).tendsto_iff hu).mp
(by simpa only [mul_one] using continuous_mul.tendsto ((1, 1) : G × G)),
simp only [and_self, mem_prod, and_imp, prod.forall, exists_true_left, prod.exists,
forall_true_left] at this,
have event_mul : ∀ n : ℕ, ∀ᶠ m in at_top, u m * u m ⊆ u n,
{ intros n,
rcases this n with ⟨j, k, h⟩,
refine at_top_basis.eventually_iff.mpr ⟨max j k, true.intro, λ m hm, _⟩,
rintro - ⟨a, b, ha, hb, rfl⟩,
exact h a b (u_anti ((le_max_left _ _).trans hm) ha) (u_anti ((le_max_right _ _).trans hm) hb)},
obtain ⟨φ, -, hφ, φ_anti_basis⟩ := has_antitone_basis.subbasis_with_rel ⟨hu, u_anti⟩ event_mul,
exact ⟨u ∘ φ, φ_anti_basis, λ n, hφ n.lt_succ_self⟩,
end
include n
/-- In a first countable topological group `G` with normal subgroup `N`, `1 : G ⧸ N` has a
countable neighborhood basis. -/
@[to_additive "In a first countable topological additive group `G` with normal additive subgroup
`N`, `0 : G ⧸ N` has a countable neighborhood basis."]
instance quotient_group.nhds_one_is_countably_generated : (𝓝 (1 : G ⧸ N)).is_countably_generated :=
(quotient_group.nhds_eq N 1).symm ▸ map.is_countably_generated _ _
end quotient_topological_group
/-- A typeclass saying that `λ p : G × G, p.1 - p.2` is a continuous function. This property
automatically holds for topological additive groups but it also holds, e.g., for `ℝ≥0`. -/
class has_continuous_sub (G : Type*) [topological_space G] [has_sub G] : Prop :=
(continuous_sub : continuous (λ p : G × G, p.1 - p.2))
/-- A typeclass saying that `λ p : G × G, p.1 / p.2` is a continuous function. This property
automatically holds for topological groups. Lemmas using this class have primes.
The unprimed version is for `group_with_zero`. -/
@[to_additive]
class has_continuous_div (G : Type*) [topological_space G] [has_div G] : Prop :=
(continuous_div' : continuous (λ p : G × G, p.1 / p.2))
@[priority 100, to_additive] -- see Note [lower instance priority]
instance topological_group.to_has_continuous_div [topological_space G] [group G]
[topological_group G] : has_continuous_div G :=
⟨by { simp only [div_eq_mul_inv], exact continuous_fst.mul continuous_snd.inv }⟩
export has_continuous_sub (continuous_sub)
export has_continuous_div (continuous_div')
section has_continuous_div
variables [topological_space G] [has_div G] [has_continuous_div G]
@[to_additive sub]
lemma filter.tendsto.div' {f g : α → G} {l : filter α} {a b : G} (hf : tendsto f l (𝓝 a))
(hg : tendsto g l (𝓝 b)) : tendsto (λ x, f x / g x) l (𝓝 (a / b)) :=
(continuous_div'.tendsto (a, b)).comp (hf.prod_mk_nhds hg)
@[to_additive const_sub]
lemma filter.tendsto.const_div' (b : G) {c : G} {f : α → G} {l : filter α}
(h : tendsto f l (𝓝 c)) : tendsto (λ k : α, b / f k) l (𝓝 (b / c)) :=
tendsto_const_nhds.div' h
@[to_additive sub_const]
lemma filter.tendsto.div_const' (b : G) {c : G} {f : α → G} {l : filter α}
(h : tendsto f l (𝓝 c)) : tendsto (λ k : α, f k / b) l (𝓝 (c / b)) :=
h.div' tendsto_const_nhds
variables [topological_space α] {f g : α → G} {s : set α} {x : α}
@[continuity, to_additive sub] lemma continuous.div' (hf : continuous f) (hg : continuous g) :
continuous (λ x, f x / g x) :=
continuous_div'.comp (hf.prod_mk hg : _)
@[to_additive continuous_sub_left]
lemma continuous_div_left' (a : G) : continuous (λ b : G, a / b) :=
continuous_const.div' continuous_id
@[to_additive continuous_sub_right]
lemma continuous_div_right' (a : G) : continuous (λ b : G, b / a) :=
continuous_id.div' continuous_const
@[to_additive sub]
lemma continuous_at.div' {f g : α → G} {x : α} (hf : continuous_at f x) (hg : continuous_at g x) :
continuous_at (λx, f x / g x) x :=
hf.div' hg
@[to_additive sub]
lemma continuous_within_at.div' (hf : continuous_within_at f s x)
(hg : continuous_within_at g s x) :
continuous_within_at (λ x, f x / g x) s x :=
hf.div' hg
@[to_additive sub]
lemma continuous_on.div' (hf : continuous_on f s) (hg : continuous_on g s) :
continuous_on (λx, f x / g x) s :=
λ x hx, (hf x hx).div' (hg x hx)
end has_continuous_div
section div_in_topological_group
variables [group G] [topological_space G] [topological_group G]
/-- A version of `homeomorph.mul_left a b⁻¹` that is defeq to `a / b`. -/
@[to_additive /-" A version of `homeomorph.add_left a (-b)` that is defeq to `a - b`. "-/,
simps {simp_rhs := tt}]
def homeomorph.div_left (x : G) : G ≃ₜ G :=
{ continuous_to_fun := continuous_const.div' continuous_id,
continuous_inv_fun := continuous_inv.mul continuous_const,
.. equiv.div_left x }
@[to_additive] lemma is_open_map_div_left (a : G) : is_open_map ((/) a) :=
(homeomorph.div_left _).is_open_map
@[to_additive] lemma is_closed_map_div_left (a : G) : is_closed_map ((/) a) :=
(homeomorph.div_left _).is_closed_map
/-- A version of `homeomorph.mul_right a⁻¹ b` that is defeq to `b / a`. -/
@[to_additive /-" A version of `homeomorph.add_right (-a) b` that is defeq to `b - a`. "-/,
simps {simp_rhs := tt}]
def homeomorph.div_right (x : G) : G ≃ₜ G :=
{ continuous_to_fun := continuous_id.div' continuous_const,
continuous_inv_fun := continuous_id.mul continuous_const,
.. equiv.div_right x }
@[to_additive]
lemma is_open_map_div_right (a : G) : is_open_map (λ x, x / a) :=
(homeomorph.div_right a).is_open_map
@[to_additive]
lemma is_closed_map_div_right (a : G) : is_closed_map (λ x, x / a) :=
(homeomorph.div_right a).is_closed_map
@[to_additive]
lemma tendsto_div_nhds_one_iff
{α : Type*} {l : filter α} {x : G} {u : α → G} :
tendsto (λ n, u n / x) l (𝓝 1) ↔ tendsto u l (𝓝 x) :=
begin
have A : tendsto (λ (n : α), x) l (𝓝 x) := tendsto_const_nhds,
exact ⟨λ h, by simpa using h.mul A, λ h, by simpa using h.div' A⟩
end
@[to_additive] lemma nhds_translation_div (x : G) : comap (/ x) (𝓝 1) = 𝓝 x :=
by simpa only [div_eq_mul_inv] using nhds_translation_mul_inv x
end div_in_topological_group
/-!
### Topological operations on pointwise sums and products
A few results about interior and closure of the pointwise addition/multiplication of sets in groups
with continuous addition/multiplication. See also `submonoid.top_closure_mul_self_eq` in
`topology.algebra.monoid`.
-/
section has_continuous_const_smul
variables [topological_space β] [group α] [mul_action α β]
[has_continuous_const_smul α β] {s : set α} {t : set β}
@[to_additive] lemma is_open.smul_left (ht : is_open t) : is_open (s • t) :=
by { rw ←bUnion_smul_set, exact is_open_bUnion (λ a _, ht.smul _) }
@[to_additive] lemma subset_interior_smul_right : s • interior t ⊆ interior (s • t) :=
interior_maximal (set.smul_subset_smul_left interior_subset) is_open_interior.smul_left
variables [topological_space α]
@[to_additive] lemma subset_interior_smul : interior s • interior t ⊆ interior (s • t) :=
(set.smul_subset_smul_right interior_subset).trans subset_interior_smul_right
end has_continuous_const_smul
section has_continuous_const_smul
variables [topological_space α] [group α] [has_continuous_const_smul α α] {s t : set α}
@[to_additive] lemma is_open.mul_left : is_open t → is_open (s * t) := is_open.smul_left
@[to_additive] lemma subset_interior_mul_right : s * interior t ⊆ interior (s * t) :=
subset_interior_smul_right
@[to_additive] lemma subset_interior_mul : interior s * interior t ⊆ interior (s * t) :=
subset_interior_smul
end has_continuous_const_smul
section has_continuous_const_smul_op
variables [topological_space α] [group α] [has_continuous_const_smul αᵐᵒᵖ α] {s t : set α}
@[to_additive] lemma is_open.mul_right (hs : is_open s) : is_open (s * t) :=
by { rw ←bUnion_op_smul_set, exact is_open_bUnion (λ a _, hs.smul _) }
@[to_additive] lemma subset_interior_mul_left : interior s * t ⊆ interior (s * t) :=
interior_maximal (set.mul_subset_mul_right interior_subset) is_open_interior.mul_right
@[to_additive] lemma subset_interior_mul' : interior s * interior t ⊆ interior (s * t) :=
(set.mul_subset_mul_left interior_subset).trans subset_interior_mul_left
end has_continuous_const_smul_op
section topological_group
variables [topological_space α] [group α] [topological_group α] {s t : set α}
@[to_additive] lemma is_open.div_left (ht : is_open t) : is_open (s / t) :=
by { rw ←Union_div_left_image, exact is_open_bUnion (λ a ha, is_open_map_div_left a t ht) }
@[to_additive] lemma is_open.div_right (hs : is_open s) : is_open (s / t) :=
by { rw ←Union_div_right_image, exact is_open_bUnion (λ a ha, is_open_map_div_right a s hs) }
@[to_additive] lemma subset_interior_div_left : interior s / t ⊆ interior (s / t) :=
interior_maximal (div_subset_div_right interior_subset) is_open_interior.div_right
@[to_additive] lemma subset_interior_div_right : s / interior t ⊆ interior (s / t) :=
interior_maximal (div_subset_div_left interior_subset) is_open_interior.div_left
@[to_additive] lemma subset_interior_div : interior s / interior t ⊆ interior (s / t) :=
(div_subset_div_left interior_subset).trans subset_interior_div_left
@[to_additive] lemma is_open.mul_closure (hs : is_open s) (t : set α) : s * closure t = s * t :=
begin
refine (mul_subset_iff.2 $ λ a ha b hb, _).antisymm (mul_subset_mul_left subset_closure),
rw mem_closure_iff at hb,
have hbU : b ∈ s⁻¹ * {a * b} := ⟨a⁻¹, a * b, set.inv_mem_inv.2 ha, rfl, inv_mul_cancel_left _ _⟩,
obtain ⟨_, ⟨c, d, hc, (rfl : d = _), rfl⟩, hcs⟩ := hb _ hs.inv.mul_right hbU,
exact ⟨c⁻¹, _, hc, hcs, inv_mul_cancel_left _ _⟩,
end
@[to_additive] lemma is_open.closure_mul (ht : is_open t) (s : set α) : closure s * t = s * t :=
by rw [←inv_inv (closure s * t), mul_inv_rev, inv_closure, ht.inv.mul_closure, mul_inv_rev, inv_inv,
inv_inv]
@[to_additive] lemma is_open.div_closure (hs : is_open s) (t : set α) : s / closure t = s / t :=