mathlib documentation

topology.algebra.group

Theory of topological groups #

This file defines the following typeclasses:

There is an instance deducing has_continuous_sub from topological_group but we use a separate typeclass because, e.g., and ℝ≥0 have continuous subtraction but are not additive groups.

We also define homeomorph versions of several equivs: homeomorph.mul_left, homeomorph.mul_right, homeomorph.inv, and prove a few facts about neighbourhood filters in groups.

Tags #

topological space, group, topological group

Groups with continuous multiplication #

In this section we prove a few statements about groups with continuous (*).

@[protected]
def homeomorph.add_left {G : Type w} [topological_space G] [add_group G] [has_continuous_add G] (a : G) :
G ≃ₜ G

Addition from the left in a topological additive group as a homeomorphism.

Equations
@[protected]
def homeomorph.mul_left {G : Type w} [topological_space G] [group G] [has_continuous_mul G] (a : G) :
G ≃ₜ G

Multiplication from the left in a topological group as a homeomorphism.

Equations
theorem is_open_map_add_left {G : Type w} [topological_space G] [add_group G] [has_continuous_add G] (a : G) :
is_open_map (λ (x : G), a + x)
theorem is_open_map_mul_left {G : Type w} [topological_space G] [group G] [has_continuous_mul G] (a : G) :
is_open_map (λ (x : G), a * x)
theorem is_closed_map_mul_left {G : Type w} [topological_space G] [group G] [has_continuous_mul G] (a : G) :
is_closed_map (λ (x : G), a * x)
theorem is_closed_map_add_left {G : Type w} [topological_space G] [add_group G] [has_continuous_add G] (a : G) :
is_closed_map (λ (x : G), a + x)
@[protected]
def homeomorph.add_right {G : Type w} [topological_space G] [add_group G] [has_continuous_add G] (a : G) :
G ≃ₜ G

Addition from the right in a topological additive group as a homeomorphism.

Equations
@[protected]
def homeomorph.mul_right {G : Type w} [topological_space G] [group G] [has_continuous_mul G] (a : G) :
G ≃ₜ G

Multiplication from the right in a topological group as a homeomorphism.

Equations
@[simp]
theorem homeomorph.coe_mul_right {G : Type w} [topological_space G] [group G] [has_continuous_mul G] (a : G) :
(homeomorph.mul_right a) = λ (g : G), g * a
@[simp]
theorem homeomorph.coe_add_right {G : Type w} [topological_space G] [add_group G] [has_continuous_add G] (a : G) :
(homeomorph.add_right a) = λ (g : G), g + a
theorem is_open_map_mul_right {G : Type w} [topological_space G] [group G] [has_continuous_mul G] (a : G) :
is_open_map (λ (x : G), x * a)
theorem is_open_map_add_right {G : Type w} [topological_space G] [add_group G] [has_continuous_add G] (a : G) :
is_open_map (λ (x : G), x + a)
theorem is_closed_map_mul_right {G : Type w} [topological_space G] [group G] [has_continuous_mul G] (a : G) :
is_closed_map (λ (x : G), x * a)
theorem is_closed_map_add_right {G : Type w} [topological_space G] [add_group G] [has_continuous_add G] (a : G) :
is_closed_map (λ (x : G), x + a)
theorem is_open_map_sub_right {G : Type w} [topological_space G] [add_group G] [has_continuous_add G] (a : G) :
is_open_map (λ (x : G), x - a)
theorem is_open_map_div_right {G : Type w} [topological_space G] [group G] [has_continuous_mul G] (a : G) :
is_open_map (λ (x : G), x / a)
theorem is_closed_map_sub_right {G : Type w} [topological_space G] [add_group G] [has_continuous_add G] (a : G) :
is_closed_map (λ (x : G), x - a)
theorem is_closed_map_div_right {G : Type w} [topological_space G] [group G] [has_continuous_mul G] (a : G) :
is_closed_map (λ (x : G), x / a)

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.

theorem is_open.mul_left {α : Type u} [topological_space α] [group α] [has_continuous_mul α] {s t : set α} (ht : is_open t) :
is_open (s * t)
theorem is_open.add_left {α : Type u} [topological_space α] [add_group α] [has_continuous_add α] {s t : set α} (ht : is_open t) :
is_open (s + t)
theorem is_open.mul_right {α : Type u} [topological_space α] [group α] [has_continuous_mul α] {s t : set α} (hs : is_open s) :
is_open (s * t)
theorem is_open.add_right {α : Type u} [topological_space α] [add_group α] [has_continuous_add α] {s t : set α} (hs : is_open s) :
is_open (s + t)
theorem subset_interior_mul_left {α : Type u} [topological_space α] [group α] [has_continuous_mul α] {s t : set α} :
(interior s) * t interior (s * t)
theorem subset_interior_add_left {α : Type u} [topological_space α] [add_group α] [has_continuous_add α] {s t : set α} :
interior s + t interior (s + t)
theorem subset_interior_add_right {α : Type u} [topological_space α] [add_group α] [has_continuous_add α] {s t : set α} :
s + interior t interior (s + t)
theorem subset_interior_mul_right {α : Type u} [topological_space α] [group α] [has_continuous_mul α] {s t : set α} :
s * interior t interior (s * t)
theorem subset_interior_add {α : Type u} [topological_space α] [add_group α] [has_continuous_add α] {s t : set α} :
theorem subset_interior_mul {α : Type u} [topological_space α] [group α] [has_continuous_mul α] {s t : set α} :

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.

theorem filter.tendsto.inv {α : Type u} {G : Type w} [topological_space G] [group G] [topological_group G] {f : α → G} {l : filter α} {y : G} (h : filter.tendsto f l (𝓝 y)) :
filter.tendsto (λ (x : α), (f x)⁻¹) l (𝓝 y⁻¹)

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'.

theorem filter.tendsto.neg {α : Type u} {G : Type w} [topological_space G] [add_group G] [topological_add_group G] {f : α → G} {l : filter α} {y : G} (h : filter.tendsto f l (𝓝 y)) :
filter.tendsto (λ (x : α), -f x) l (𝓝 (-y))
@[continuity]
theorem continuous.inv {α : Type u} {G : Type w} [topological_space G] [group G] [topological_group G] [topological_space α] {f : α → G} (hf : continuous f) :
continuous (λ (x : α), (f x)⁻¹)
@[continuity]
theorem continuous.neg {α : Type u} {G : Type w} [topological_space G] [add_group G] [topological_add_group G] [topological_space α] {f : α → G} (hf : continuous f) :
continuous (λ (x : α), -f x)
theorem continuous_at.neg {α : Type u} {G : Type w} [topological_space G] [add_group G] [topological_add_group G] [topological_space α] {f : α → G} {x : α} (hf : continuous_at f x) :
continuous_at (λ (x : α), -f x) x
theorem continuous_at.inv {α : Type u} {G : Type w} [topological_space G] [group G] [topological_group G] [topological_space α] {f : α → G} {x : α} (hf : continuous_at f x) :
continuous_at (λ (x : α), (f x)⁻¹) x
theorem continuous_on.inv {α : Type u} {G : Type w} [topological_space G] [group G] [topological_group G] [topological_space α] {f : α → G} {s : set α} (hf : continuous_on f s) :
continuous_on (λ (x : α), (f x)⁻¹) s
theorem continuous_on.neg {α : Type u} {G : Type w} [topological_space G] [add_group G] [topological_add_group G] [topological_space α] {f : α → G} {s : set α} (hf : continuous_on f s) :
continuous_on (λ (x : α), -f x) s
theorem continuous_within_at.inv {α : Type u} {G : Type w} [topological_space G] [group G] [topological_group G] [topological_space α] {f : α → G} {s : set α} {x : α} (hf : continuous_within_at f s x) :
continuous_within_at (λ (x : α), (f x)⁻¹) s x
theorem continuous_within_at.neg {α : Type u} {G : Type w} [topological_space G] [add_group G] [topological_add_group G] [topological_space α] {f : α → G} {s : set α} {x : α} (hf : continuous_within_at f s x) :
continuous_within_at (λ (x : α), -f x) s x
@[protected, instance]
@[protected, instance]
def pi.topological_add_group {β : Type v} {C : β → Type u_1} [Π (b : β), topological_space (C b)] [Π (b : β), add_group (C b)] [∀ (b : β), topological_add_group (C b)] :
topological_add_group (Π (b : β), C b)
@[protected, instance]
def pi.topological_group {β : Type v} {C : β → Type u_1} [Π (b : β), topological_space (C b)] [Π (b : β), group (C b)] [∀ (b : β), topological_group (C b)] :
topological_group (Π (b : β), C b)
@[protected]

Negation in a topological group as a homeomorphism.

Equations
@[protected]
def homeomorph.inv (G : Type w) [topological_space G] [group G] [topological_group G] :
G ≃ₜ G

Inversion in a topological group as a homeomorphism.

Equations
@[protected]

The map (x, y) ↦ (x, xy) as a homeomorphism. This is a shear mapping.

Equations
@[protected]

The map (x, y) ↦ (x, x + y) as a homeomorphism. This is a shear mapping.

Equations
@[simp]
theorem homeomorph.shear_mul_right_coe (G : Type w) [topological_space G] [group G] [topological_group G] :
(homeomorph.shear_mul_right G) = λ (z : G × G), (z.fst, (z.fst) * z.snd)
@[simp]
@[simp]
theorem is_open.inv {G : Type w} [topological_space G] [group G] [topological_group G] {s : set G} (hs : is_open s) :
theorem is_open.neg {G : Type w} [topological_space G] [add_group G] [topological_add_group G] {s : set G} (hs : is_open s) :
theorem is_closed.neg {G : Type w} [topological_space G] [add_group G] [topological_add_group G] {s : set G} (hs : is_closed s) :
theorem is_closed.inv {G : Type w} [topological_space G] [group G] [topological_group G] {s : set G} (hs : is_closed s) :
@[protected, instance]
theorem neg_closure {G : Type w} [topological_space G] [add_group G] [topological_add_group G] (s : set G) :
theorem inv_closure {G : Type w} [topological_space G] [group G] [topological_group G] (s : set G) :

The (topological-space) closure of a subgroup of a space M with has_continuous_mul is itself a subgroup.

Equations

The (topological-space) closure of an additive subgroup of a space M with has_continuous_add is itself an additive subgroup.

Equations
theorem exists_nhds_half_neg {G : Type w} [topological_space G] [add_group G] [topological_add_group G] {s : set G} (hs : s 𝓝 0) :
∃ (V : set G) (H : V 𝓝 0), ∀ (v : G), v V∀ (w : G), w Vv - w s
theorem exists_nhds_split_inv {G : Type w} [topological_space G] [group G] [topological_group G] {s : set G} (hs : s 𝓝 1) :
∃ (V : set G) (H : V 𝓝 1), ∀ (v : G), v V∀ (w : G), w Vv / w s
theorem nhds_translation_add_neg {G : Type w} [topological_space G] [add_group G] [topological_add_group G] (x : G) :
filter.comap (λ (y : G), y + -x) (𝓝 0) = 𝓝 x
theorem nhds_translation_mul_inv {G : Type w} [topological_space G] [group G] [topological_group G] (x : G) :
filter.comap (λ (y : G), y * x⁻¹) (𝓝 1) = 𝓝 x
@[simp]
theorem map_add_left_nhds {G : Type w} [topological_space G] [add_group G] [topological_add_group G] (x y : G) :
@[simp]
theorem map_mul_left_nhds {G : Type w} [topological_space G] [group G] [topological_group G] (x y : G) :
theorem map_mul_left_nhds_one {G : Type w} [topological_space G] [group G] [topological_group G] (x : G) :
theorem topological_group.ext {G : Type u_1} [group G] {t t' : topological_space G} (tg : topological_group G) (tg' : topological_group G) (h : 𝓝 1 = 𝓝 1) :
t = t'
theorem topological_add_group.ext {G : Type u_1} [add_group G] {t t' : topological_space G} (tg : topological_add_group G) (tg' : topological_add_group G) (h : 𝓝 0 = 𝓝 0) :
t = t'
theorem topological_group.of_nhds_aux {G : Type u_1} [group G] [topological_space G] (hinv : filter.tendsto (λ (x : G), x⁻¹) (𝓝 1) (𝓝 1)) (hleft : ∀ (x₀ : G), 𝓝 x₀ = filter.map (λ (x : G), x₀ * x) (𝓝 1)) (hconj : ∀ (x₀ : G), filter.map (λ (x : G), (x₀ * x) * x₀⁻¹) (𝓝 1) 𝓝 1) :
continuous (λ (x : G), x⁻¹)
theorem topological_add_group.of_nhds_aux {G : Type u_1} [add_group G] [topological_space G] (hinv : filter.tendsto (λ (x : G), -x) (𝓝 0) (𝓝 0)) (hleft : ∀ (x₀ : G), 𝓝 x₀ = filter.map (λ (x : G), x₀ + x) (𝓝 0)) (hconj : ∀ (x₀ : G), filter.map (λ (x : G), x₀ + x + -x₀) (𝓝 0) 𝓝 0) :
continuous (λ (x : G), -x)
theorem topological_group.of_nhds_one' {G : Type u} [group G] [topological_space G] (hmul : filter.tendsto (function.uncurry has_mul.mul) (𝓝 1 ×ᶠ 𝓝 1) (𝓝 1)) (hinv : filter.tendsto (λ (x : G), x⁻¹) (𝓝 1) (𝓝 1)) (hleft : ∀ (x₀ : G), 𝓝 x₀ = filter.map (λ (x : G), x₀ * x) (𝓝 1)) (hright : ∀ (x₀ : G), 𝓝 x₀ = filter.map (λ (x : G), x * x₀) (𝓝 1)) :
theorem topological_add_group.of_nhds_zero' {G : Type u} [add_group G] [topological_space G] (hmul : filter.tendsto (function.uncurry has_add.add) (𝓝 0 ×ᶠ 𝓝 0) (𝓝 0)) (hinv : filter.tendsto (λ (x : G), -x) (𝓝 0) (𝓝 0)) (hleft : ∀ (x₀ : G), 𝓝 x₀ = filter.map (λ (x : G), x₀ + x) (𝓝 0)) (hright : ∀ (x₀ : G), 𝓝 x₀ = filter.map (λ (x : G), x + x₀) (𝓝 0)) :
theorem topological_add_group.of_nhds_zero {G : Type u} [add_group G] [topological_space G] (hmul : filter.tendsto (function.uncurry has_add.add) (𝓝 0 ×ᶠ 𝓝 0) (𝓝 0)) (hinv : filter.tendsto (λ (x : G), -x) (𝓝 0) (𝓝 0)) (hleft : ∀ (x₀ : G), 𝓝 x₀ = filter.map (λ (x : G), x₀ + x) (𝓝 0)) (hconj : ∀ (x₀ : G), filter.tendsto (λ (x : G), x₀ + x + -x₀) (𝓝 0) (𝓝 0)) :
theorem topological_group.of_nhds_one {G : Type u} [group G] [topological_space G] (hmul : filter.tendsto (function.uncurry has_mul.mul) (𝓝 1 ×ᶠ 𝓝 1) (𝓝 1)) (hinv : filter.tendsto (λ (x : G), x⁻¹) (𝓝 1) (𝓝 1)) (hleft : ∀ (x₀ : G), 𝓝 x₀ = filter.map (λ (x : G), x₀ * x) (𝓝 1)) (hconj : ∀ (x₀ : G), filter.tendsto (λ (x : G), (x₀ * x) * x₀⁻¹) (𝓝 1) (𝓝 1)) :
theorem topological_add_group.of_comm_of_nhds_zero {G : Type u} [add_comm_group G] [topological_space G] (hmul : filter.tendsto (function.uncurry has_add.add) (𝓝 0 ×ᶠ 𝓝 0) (𝓝 0)) (hinv : filter.tendsto (λ (x : G), -x) (𝓝 0) (𝓝 0)) (hleft : ∀ (x₀ : G), 𝓝 x₀ = filter.map (λ (x : G), x₀ + x) (𝓝 0)) :
theorem topological_group.of_comm_of_nhds_one {G : Type u} [comm_group G] [topological_space G] (hmul : filter.tendsto (function.uncurry has_mul.mul) (𝓝 1 ×ᶠ 𝓝 1) (𝓝 1)) (hinv : filter.tendsto (λ (x : G), x⁻¹) (𝓝 1) (𝓝 1)) (hleft : ∀ (x₀ : G), 𝓝 x₀ = filter.map (λ (x : G), x₀ * x) (𝓝 1)) :
@[protected, instance]
@[class]
structure has_continuous_sub (G : Type u_1) [topological_space G] [has_sub G] :
Prop

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.

Instances
@[class]
structure has_continuous_div (G : Type u_1) [topological_space G] [has_div G] :
Prop

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.

Instances
theorem filter.tendsto.div' {α : Type u} {G : Type w} [topological_space G] [has_div G] [has_continuous_div G] {f g : α → G} {l : filter α} {a b : G} (hf : filter.tendsto f l (𝓝 a)) (hg : filter.tendsto g l (𝓝 b)) :
filter.tendsto (λ (x : α), f x / g x) l (𝓝 (a / b))
theorem filter.tendsto.sub {α : Type u} {G : Type w} [topological_space G] [has_sub G] [has_continuous_sub G] {f g : α → G} {l : filter α} {a b : G} (hf : filter.tendsto f l (𝓝 a)) (hg : filter.tendsto g l (𝓝 b)) :
filter.tendsto (λ (x : α), f x - g x) l (𝓝 (a - b))
theorem filter.tendsto.const_sub {α : Type u} {G : Type w} [topological_space G] [has_sub G] [has_continuous_sub G] (b : G) {c : G} {f : α → G} {l : filter α} (h : filter.tendsto f l (𝓝 c)) :
filter.tendsto (λ (k : α), b - f k) l (𝓝 (b - c))
theorem filter.tendsto.const_div' {α : Type u} {G : Type w} [topological_space G] [has_div G] [has_continuous_div G] (b : G) {c : G} {f : α → G} {l : filter α} (h : filter.tendsto f l (𝓝 c)) :
filter.tendsto (λ (k : α), b / f k) l (𝓝 (b / c))
theorem filter.tendsto.div_const' {α : Type u} {G : Type w} [topological_space G] [has_div G] [has_continuous_div G] (b : G) {c : G} {f : α → G} {l : filter α} (h : filter.tendsto f l (𝓝 c)) :
filter.tendsto (λ (k : α), f k / b) l (𝓝 (c / b))
theorem filter.tendsto.sub_const {α : Type u} {G : Type w} [topological_space G] [has_sub G] [has_continuous_sub G] (b : G) {c : G} {f : α → G} {l : filter α} (h : filter.tendsto f l (𝓝 c)) :
filter.tendsto (λ (k : α), f k - b) l (𝓝 (c - b))
@[continuity]
theorem continuous.div' {α : Type u} {G : Type w} [topological_space G] [has_div G] [has_continuous_div G] [topological_space α] {f g : α → G} (hf : continuous f) (hg : continuous g) :
continuous (λ (x : α), f x / g x)
@[continuity]
theorem continuous.sub {α : Type u} {G : Type w} [topological_space G] [has_sub G] [has_continuous_sub G] [topological_space α] {f g : α → G} (hf : continuous f) (hg : continuous g) :
continuous (λ (x : α), f x - g x)
theorem continuous_sub_left {G : Type w} [topological_space G] [has_sub G] [has_continuous_sub G] (a : G) :
continuous (λ (b : G), a - b)
theorem continuous_div_left' {G : Type w} [topological_space G] [has_div G] [has_continuous_div G] (a : G) :
continuous (λ (b : G), a / b)
theorem continuous_sub_right {G : Type w} [topological_space G] [has_sub G] [has_continuous_sub G] (a : G) :
continuous (λ (b : G), b - a)
theorem continuous_div_right' {G : Type w} [topological_space G] [has_div G] [has_continuous_div G] (a : G) :
continuous (λ (b : G), b / a)
theorem continuous_at.div' {α : Type u} {G : Type w} [topological_space G] [has_div G] [has_continuous_div G] [topological_space α] {f g : α → G} {x : α} (hf : continuous_at f x) (hg : continuous_at g x) :
continuous_at (λ (x : α), f x / g x) x
theorem continuous_at.sub {α : Type u} {G : Type w} [topological_space G] [has_sub G] [has_continuous_sub G] [topological_space α] {f g : α → G} {x : α} (hf : continuous_at f x) (hg : continuous_at g x) :
continuous_at (λ (x : α), f x - g x) x
theorem continuous_within_at.sub {α : Type u} {G : Type w} [topological_space G] [has_sub G] [has_continuous_sub G] [topological_space α] {f g : α → G} {s : set α} {x : α} (hf : continuous_within_at f s x) (hg : continuous_within_at g s x) :
continuous_within_at (λ (x : α), f x - g x) s x
theorem continuous_within_at.div' {α : Type u} {G : Type w} [topological_space G] [has_div G] [has_continuous_div G] [topological_space α] {f g : α → G} {s : set α} {x : α} (hf : continuous_within_at f s x) (hg : continuous_within_at g s x) :
continuous_within_at (λ (x : α), f x / g x) s x
theorem continuous_on.sub {α : Type u} {G : Type w} [topological_space G] [has_sub G] [has_continuous_sub G] [topological_space α] {f g : α → G} {s : set α} (hf : continuous_on f s) (hg : continuous_on g s) :
continuous_on (λ (x : α), f x - g x) s
theorem continuous_on.div' {α : Type u} {G : Type w} [topological_space G] [has_div G] [has_continuous_div G] [topological_space α] {f g : α → G} {s : set α} (hf : continuous_on f s) (hg : continuous_on g s) :
continuous_on (λ (x : α), f x / g x) s
theorem nhds_translation_div {G : Type w} [topological_space G] [group G] [topological_group G] (x : G) :
filter.comap (λ (y : G), y / x) (𝓝 1) = 𝓝 x
theorem nhds_translation_sub {G : Type w} [topological_space G] [add_group G] [topological_add_group G] (x : G) :
filter.comap (λ (y : G), y - x) (𝓝 0) = 𝓝 x
@[class]
structure add_group_with_zero_nhd (G : Type u) :
Type u

additive group with a neighbourhood around 0. Only used to construct a topology and uniform space.

This is currently only available for commutative groups, but it can be extended to non-commutative groups too.

Some results about an open set containing the product of two sets in a topological group.

theorem compact_open_separated_add {G : Type w} [topological_space G] [add_group G] [topological_add_group G] {K U : set G} (hK : is_compact K) (hU : is_open U) (hKU : K U) :
∃ (V : set G), is_open V 0 V K + V U

Given a compact set K inside an open set U, there is a open neighborhood V of 0 such that K + V ⊆ U.

theorem compact_open_separated_mul {G : Type w} [topological_space G] [group G] [topological_group G] {K U : set G} (hK : is_compact K) (hU : is_open U) (hKU : K U) :
∃ (V : set G), is_open V 1 V K * V U

Given a compact set K inside an open set U, there is a open neighborhood V of 1 such that KV ⊆ U.

theorem compact_covered_by_add_left_translates {G : Type w} [topological_space G] [add_group G] [topological_add_group G] {K V : set G} (hK : is_compact K) (hV : (interior V).nonempty) :
∃ (t : finset G), K ⋃ (g : G) (H : g t), (λ (h : G), g + h) ⁻¹' V

A compact set is covered by finitely many left additive translates of a set with non-empty interior.

theorem compact_covered_by_mul_left_translates {G : Type w} [topological_space G] [group G] [topological_group G] {K V : set G} (hK : is_compact K) (hV : (interior V).nonempty) :
∃ (t : finset G), K ⋃ (g : G) (H : g t), (λ (h : G), g * h) ⁻¹' V

A compact set is covered by finitely many left multiplicative translates of a set with non-empty interior.

@[protected, instance]

Every locally compact separable topological group is σ-compact. Note: this is not true if we drop the topological group hypothesis.

Every separated topological group in which there exists a compact set with nonempty interior is locally compact.

theorem nhds_mul {G : Type w} [topological_space G] [comm_group G] [topological_group G] (x y : G) :
𝓝 (x * y) = (𝓝 x) * 𝓝 y
theorem nhds_add {G : Type w} [topological_space G] [add_comm_group G] [topological_add_group G] (x y : G) :
𝓝 (x + y) = 𝓝 x + 𝓝 y
@[simp]
theorem nhds_mul_hom_apply {G : Type w} [topological_space G] [comm_group G] [topological_group G] (a : G) :

On an additive topological group, 𝓝 : G → filter G can be promoted to an add_hom.

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@[simp]

On a topological group, 𝓝 : G → filter G can be promoted to a mul_hom.

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@[protected, instance]
@[protected, instance]
def units.homeomorph.prod_units {α : Type u} {β : Type v} [monoid α] [topological_space α] [has_continuous_mul α] [monoid β] [topological_space β] [has_continuous_mul β] :
× β)ˣ ≃ₜ αˣ × βˣ

The topological group isomorphism between the units of a product of two monoids, and the product of the units of each monoid.

Equations

Lattice of group topologies #

We define a type class group_topology α which endows a group α with a topology such that all group operations are continuous.

Group topologies on a fixed group α are ordered, by reverse inclusion. They form a complete lattice, with the discrete topology and the indiscrete topology.

Any function f : α → β induces coinduced f : topological_space α → group_topology β.

The additive version add_group_topology α and corresponding results are provided as well.

structure group_topology (α : Type u) [group α] :
Type u

A group topology on a group α is a topology for which multiplication and inversion are continuous.

structure add_group_topology (α : Type u) [add_group α] :
Type u

An additive group topology on an additive group α is a topology for which addition and negation are continuous.

theorem add_group_topology.continuous_add' {α : Type u} [add_group α] (g : add_group_topology α) :
continuous (λ (p : α × α), p.fst + p.snd)
theorem group_topology.continuous_mul' {α : Type u} [group α] (g : group_topology α) :
continuous (λ (p : α × α), (p.fst) * p.snd)

A version of the global continuous_mul suitable for dot notation.

A version of the global continuous_inv suitable for dot notation.

@[ext]
theorem group_topology.ext' {α : Type u} [group α] {f g : group_topology α} (h : f.to_topological_space.is_open = g.to_topological_space.is_open) :
f = g
@[protected, instance]

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

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@[protected, instance]
def group_topology.has_top {α : Type u} [group α] :
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@[protected, instance]
def group_topology.has_bot {α : Type u} [group α] :
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@[protected, instance]
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@[protected, instance]
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@[protected, instance]
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@[protected, instance]
def group_topology.inhabited {α : Type u} [group α] :
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@[protected, instance]

Infimum of a collection of additive group topologies

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@[simp]
theorem group_topology.to_topological_space_infi {α : Type u} [group α] {ι : Sort u_1} (s : ι → group_topology α) :
(⨅ (i : ι), s i).to_topological_space = ⨅ (i : ι), (s i).to_topological_space
@[simp]
theorem add_group_topology.to_topological_space_infi {α : Type u} [add_group α] {ι : Sort u_1} (s : ι → add_group_topology α) :
(⨅ (i : ι), s i).to_topological_space = ⨅ (i : ι), (s i).to_topological_space
@[protected, instance]

Group topologies on γ form a complete lattice, with the discrete topology and the indiscrete topology.

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

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

Equations
def add_group_topology.coinduced {α : Type u_1} {β : Type u_2} [t : topological_space α] [add_group β] (f : α → β) :

Given f : α → β and a topology on α, the coinduced additive group topology on β is the finest topology such that f is continuous and β is a topological additive group.

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def group_topology.coinduced {α : Type u_1} {β : Type u_2} [t : topological_space α] [group β] (f : α → β) :

Given f : α → β and a topology on α, the coinduced group topology on β is the finest topology such that f is continuous and β is a topological group.

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theorem group_topology.coinduced_continuous {α : Type u_1} {β : Type u_2} [t : topological_space α] [group β] (f : α → β) :
theorem add_group_topology.coinduced_continuous {α : Type u_1} {β : Type u_2} [t : topological_space α] [add_group β] (f : α → β) :