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 (*).

def homeomorph.add_left {G : Type w} [topological_space G] [add_group G] [has_continuous_add G] :
G → G ≃ₜ G

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

def homeomorph.mul_left {G : Type w} [topological_space G] [group G] [has_continuous_mul G] :
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)

def homeomorph.add_right {G : Type w} [topological_space G] [add_group G] [has_continuous_add G] :
G → G ≃ₜ G

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

def homeomorph.mul_right {G : Type w} [topological_space G] [group G] [has_continuous_mul G] :
G → G ≃ₜ G

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

Equations
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)

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.

@[class]
structure topological_group (G : Type u_1) [topological_space G] [group G] :
Prop

A topological group is a group in which the multiplication and inversion operations are continuous.

Instances
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} :
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} :
filter.tendsto f l (𝓝 y)filter.tendsto (λ (x : α), -f x) l (𝓝 (-y))

theorem continuous.inv {α : Type u} {G : Type w} [topological_space G] [group G] [topological_group G] [topological_space α] {f : α → G} :
continuous fcontinuous (λ (x : α), (f x)⁻¹)

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} :
continuous fcontinuous (λ (x : α), -f 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} :
continuous_on f scontinuous_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} :
continuous_on f scontinuous_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} :
continuous_within_at f s xcontinuous_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} :
continuous_within_at f s xcontinuous_within_at (λ (x : α), -f x) s x

@[instance]

Negation in a topological group as a homeomorphism.

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
theorem exists_nhds_half_neg {G : Type w} [topological_space G] [add_group G] [topological_add_group G] {s : set G} :
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} :
s 𝓝 1(∃ (V : set G) (H : V 𝓝 1), ∀ (v : G), v V∀ (w : G), w Vv * w⁻¹ 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

theorem topological_group.ext {G : Type u_1} [group G] {t t' : topological_space G} :
topological_group Gtopological_group G𝓝 1 = 𝓝 1t = t'

theorem topological_add_group.ext {G : Type u_1} [add_group G] {t t' : topological_space G} :

@[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
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} :
filter.tendsto f l (𝓝 a)filter.tendsto g l (𝓝 b)filter.tendsto (λ (x : α), f x - g x) l (𝓝 (a - b))

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} :
continuous fcontinuous gcontinuous (λ (x : α), f x - g 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} :
continuous_within_at f s xcontinuous_within_at g s xcontinuous_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} :
continuous_on f scontinuous_on g scontinuous_on (λ (x : α), f x - g x) s

theorem nhds_translation {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  :
Type uType 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.

theorem add_group_with_zero_nhd.exists_Z_half {G : Type w} [add_group_with_zero_nhd G] {s : set G} :
s add_group_with_zero_nhd.Z G(∃ (V : set G) (H : V add_group_with_zero_nhd.Z G), ∀ (v : G), v V∀ (w : G), w Vv + w s)

theorem add_group_with_zero_nhd.nhds_eq {G : Type w} [add_group_with_zero_nhd G] (a : G) :
𝓝 a = filter.map (λ (x : G), x + a) (add_group_with_zero_nhd.Z G)

theorem is_open.mul_left {G : Type w} [topological_space G] [group G] [topological_group G] {s t : set G} :
is_open tis_open (s * t)

theorem is_open.add_left {G : Type w} [topological_space G] [add_group G] [topological_add_group G] {s t : set G} :
is_open tis_open (s + t)

theorem is_open.mul_right {G : Type w} [topological_space G] [group G] [topological_group G] {s t : set G} :
is_open sis_open (s * t)

theorem is_open.add_right {G : Type w} [topological_space G] [add_group G] [topological_add_group G] {s t : set G} :
is_open sis_open (s + t)

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} :
is_compact Kis_open UK 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} :
is_compact Kis_open UK 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} :
is_compact K(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} :
is_compact K(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.

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

theorem nhds_is_add_hom {G : Type w} [topological_space G] [add_comm_group G] [topological_add_group G] :
is_add_hom (λ (x : G), 𝓝 x)

theorem nhds_is_mul_hom {G : Type w} [topological_space G] [comm_group G] [topological_group G] :
is_mul_hom (λ (x : G), 𝓝 x)