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/- | |
Copyright (c) 2019 Johan Commelin All rights reserved. | |
Released under Apache 2.0 license as described in the file LICENSE. | |
Authors: Johan Commelin | |
-/ | |
import topology.algebra.ring | |
import topology.algebra.filter_basis | |
import topology.sets.opens | |
/-! | |
# Open subgroups of a topological groups | |
This files builds the lattice `open_subgroup G`Β of open subgroups in a topological group `G`, | |
and its additive version `open_add_subgroup`. 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 | |
* `open_subgroup.is_closed`: An open subgroup is automatically closed. | |
* `subgroup.is_open_mono`: A subgroup containing an open subgroup is open. | |
There are also versions for additive groups, submodules and ideals. | |
* `open_subgroup.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. | |
-/ | |
open topological_space | |
open_locale topological_space | |
/-- The type of open subgroups of a topological additive group. -/ | |
@[ancestor add_subgroup] | |
structure open_add_subgroup (G : Type*) [add_group G] [topological_space G] | |
extends add_subgroup G := | |
(is_open' : is_open carrier) | |
/-- The type of open subgroups of a topological group. -/ | |
@[ancestor subgroup, to_additive] | |
structure open_subgroup (G : Type*) [group G] [topological_space G] extends subgroup G := | |
(is_open' : is_open carrier) | |
/-- Reinterpret an `open_subgroup` as a `subgroup`. -/ | |
add_decl_doc open_subgroup.to_subgroup | |
/-- Reinterpret an `open_add_subgroup` as an `add_subgroup`. -/ | |
add_decl_doc open_add_subgroup.to_add_subgroup | |
namespace open_subgroup | |
open function topological_space | |
variables {G : Type*} [group G] [topological_space G] | |
variables {U V : open_subgroup G} {g : G} | |
@[to_additive] | |
instance has_coe_set : has_coe_t (open_subgroup G) (set G) := β¨Ξ» U, U.1β© | |
@[to_additive] | |
instance : has_mem G (open_subgroup G) := β¨Ξ» g U, g β (U : set G)β© | |
@[to_additive] | |
instance has_coe_subgroup : has_coe_t (open_subgroup G) (subgroup G) := β¨to_subgroupβ© | |
@[to_additive] | |
instance has_coe_opens : has_coe_t (open_subgroup G) (opens G) := β¨Ξ» U, β¨U, U.is_open'β©β© | |
@[simp, norm_cast, to_additive] lemma mem_coe : g β (U : set G) β g β U := iff.rfl | |
@[simp, norm_cast, to_additive] lemma mem_coe_opens : g β (U : opens G) β g β U := iff.rfl | |
@[simp, norm_cast, to_additive] | |
lemma mem_coe_subgroup : g β (U : subgroup G) β g β U := iff.rfl | |
@[to_additive] lemma coe_injective : injective (coe : open_subgroup G β set G) := | |
by { rintros β¨β¨β©β© β¨β¨β©β© β¨hβ©, congr, } | |
@[ext, to_additive] | |
lemma ext (h : β x, x β U β x β V) : (U = V) := coe_injective $ set.ext h | |
@[to_additive] | |
lemma ext_iff : (U = V) β (β x, x β U β x β V) := β¨Ξ» h x, h βΈ iff.rfl, extβ© | |
variable (U) | |
@[to_additive] | |
protected lemma is_open : is_open (U : set G) := U.is_open' | |
@[to_additive] | |
protected lemma one_mem : (1 : G) β U := U.one_mem' | |
@[to_additive] | |
protected lemma inv_mem {g : G} (h : g β U) : gβ»ΒΉ β U := U.inv_mem' h | |
@[to_additive] | |
protected lemma mul_mem {gβ gβ : G} (hβ : gβ β U) (hβ : gβ β U) : gβ * gβ β U := U.mul_mem' hβ hβ | |
@[to_additive] | |
lemma mem_nhds_one : (U : set G) β π (1 : G) := | |
is_open.mem_nhds U.is_open U.one_mem | |
variable {U} | |
@[to_additive] | |
instance : has_top (open_subgroup G) := β¨{ is_open' := is_open_univ, .. (β€ : subgroup G) }β© | |
@[to_additive] | |
instance : inhabited (open_subgroup G) := β¨β€β© | |
@[to_additive] | |
lemma is_closed [has_continuous_mul G] (U : open_subgroup G) : is_closed (U : set G) := | |
begin | |
apply is_open_compl_iff.1, | |
refine is_open_iff_forall_mem_open.2 (Ξ» x hx, β¨(Ξ» y, y * xβ»ΒΉ) β»ΒΉ' U, _, _, _β©), | |
{ intros u hux, | |
simp only [set.mem_preimage, set.mem_compl_iff, mem_coe] at hux hx β’, | |
refine mt (Ξ» hu, _) hx, | |
convert U.mul_mem (U.inv_mem hux) hu, | |
simp }, | |
{ exact U.is_open.preimage (continuous_mul_right _) }, | |
{ simp [U.one_mem] } | |
end | |
section | |
variables {H : Type*} [group H] [topological_space H] | |
/-- The product of two open subgroups as an open subgroup of the product group. -/ | |
@[to_additive "The product of two open subgroups as an open subgroup of the product group."] | |
def prod (U : open_subgroup G) (V : open_subgroup H) : open_subgroup (G Γ H) := | |
{ carrier := (U : set G) ΓΛ’ (V : set H), | |
is_open' := U.is_open.prod V.is_open, | |
.. (U : subgroup G).prod (V : subgroup H) } | |
end | |
@[to_additive] | |
instance : partial_order (open_subgroup G) := | |
{ le := Ξ» U V, β β¦xβ¦, x β U β x β V, | |
.. partial_order.lift (coe : open_subgroup G β set G) coe_injective } | |
@[to_additive] | |
instance : semilattice_inf (open_subgroup G) := | |
{ inf := Ξ» U V, { is_open' := is_open.inter U.is_open V.is_open, .. (U : subgroup G) β V }, | |
inf_le_left := Ξ» U V, set.inter_subset_left _ _, | |
inf_le_right := Ξ» U V, set.inter_subset_right _ _, | |
le_inf := Ξ» U V W hV hW, set.subset_inter hV hW, | |
..open_subgroup.partial_order } | |
@[to_additive] | |
instance : order_top (open_subgroup G) := | |
{ top := β€, | |
le_top := Ξ» U, set.subset_univ _ } | |
@[simp, norm_cast, to_additive] lemma coe_inf : (β(U β V) : set G) = (U : set G) β© V := rfl | |
@[simp, norm_cast, to_additive] lemma coe_subset : (U : set G) β V β U β€ V := iff.rfl | |
@[simp, norm_cast, to_additive] lemma coe_subgroup_le : | |
(U : subgroup G) β€ (V : subgroup G) β U β€ V := iff.rfl | |
variables {N : Type*} [group N] [topological_space N] | |
/-- The preimage of an `open_subgroup` along a continuous `monoid` homomorphism | |
is an `open_subgroup`. -/ | |
@[to_additive "The preimage of an `open_add_subgroup` along a continuous `add_monoid` homomorphism | |
is an `open_add_subgroup`."] | |
def comap (f : G β* N) | |
(hf : continuous f) (H : open_subgroup N) : open_subgroup G := | |
{ is_open' := H.is_open.preimage hf, | |
.. (H : subgroup N).comap f } | |
@[simp, to_additive] | |
lemma coe_comap (H : open_subgroup N) (f : G β* N) (hf : continuous f) : | |
(H.comap f hf : set G) = f β»ΒΉ' H := rfl | |
@[simp, to_additive] | |
lemma mem_comap {H : open_subgroup N} {f : G β* N} {hf : continuous f} {x : G} : | |
x β H.comap f hf β f x β H := iff.rfl | |
@[to_additive] | |
lemma comap_comap {P : Type*} [group P] [topological_space P] | |
(K : open_subgroup P) (fβ : N β* P) (hfβ : continuous fβ) (fβ : G β* N) (hfβ : continuous fβ) : | |
(K.comap fβ hfβ).comap fβ hfβ = K.comap (fβ.comp fβ) (hfβ.comp hfβ) := | |
rfl | |
end open_subgroup | |
namespace subgroup | |
variables {G : Type*} [group G] [topological_space G] [has_continuous_mul G] (H : subgroup G) | |
@[to_additive] | |
lemma is_open_of_mem_nhds {g : G} (hg : (H : set G) β π g) : | |
is_open (H : set G) := | |
begin | |
simp only [is_open_iff_mem_nhds, set_like.mem_coe] at hg β’, | |
intros x hx, | |
have : filter.tendsto (Ξ» y, y * (xβ»ΒΉ * g)) (π x) (π $ x * (xβ»ΒΉ * g)) := | |
(continuous_id.mul continuous_const).tendsto _, | |
rw [mul_inv_cancel_left] at this, | |
have := filter.mem_map'.1 (this hg), | |
replace hg : g β H := set_like.mem_coe.1 (mem_of_mem_nhds hg), | |
simp only [set_like.mem_coe, H.mul_mem_cancel_right (H.mul_mem (H.inv_mem hx) hg)] at this, | |
exact this | |
end | |
@[to_additive] | |
lemma is_open_of_open_subgroup {U : open_subgroup G} (h : U.1 β€ H) : | |
is_open (H : set G) := | |
H.is_open_of_mem_nhds (filter.mem_of_superset U.mem_nhds_one h) | |
/-- If a subgroup of a topological group has `1` in its interior, then it is open. -/ | |
@[to_additive "If a subgroup of an additive topological group has `0` in its interior, then it is | |
open."] | |
lemma is_open_of_one_mem_interior {G : Type*} [group G] [topological_space G] | |
[topological_group G] {H : subgroup G} (h_1_int : (1 : G) β interior (H : set G)) : | |
is_open (H : set G) := | |
begin | |
have h : π 1 β€ filter.principal (H : set G) := | |
nhds_le_of_le h_1_int (is_open_interior) (filter.principal_mono.2 interior_subset), | |
rw is_open_iff_nhds, | |
intros g hg, | |
rw (show π g = filter.map β(homeomorph.mul_left g) (π 1), by simp), | |
convert filter.map_mono h, | |
simp only [homeomorph.coe_mul_left, filter.map_principal, set.image_mul_left, | |
filter.principal_eq_iff_eq], | |
ext, | |
simp [H.mul_mem_cancel_left (H.inv_mem hg)], | |
end | |
@[to_additive] | |
lemma is_open_mono {Hβ Hβ : subgroup G} (h : Hβ β€ Hβ) (hβ : is_open (Hβ : set G)) : | |
is_open (Hβ : set G) := | |
@is_open_of_open_subgroup _ _ _ _ Hβ { is_open' := hβ, .. Hβ } h | |
end subgroup | |
namespace open_subgroup | |
variables {G : Type*} [group G] [topological_space G] [has_continuous_mul G] | |
@[to_additive] | |
instance : semilattice_sup (open_subgroup G) := | |
{ sup := Ξ» U V, | |
{ is_open' := show is_open (((U : subgroup G) β V : subgroup G) : set G), | |
from subgroup.is_open_mono le_sup_left U.is_open, | |
.. ((U : subgroup G) β V) }, | |
le_sup_left := Ξ» U V, coe_subgroup_le.1 le_sup_left, | |
le_sup_right := Ξ» U V, coe_subgroup_le.1 le_sup_right, | |
sup_le := Ξ» U V W hU hV, coe_subgroup_le.1 (sup_le hU hV), | |
..open_subgroup.semilattice_inf } | |
@[to_additive] | |
instance : lattice (open_subgroup G) := | |
{ ..open_subgroup.semilattice_sup, ..open_subgroup.semilattice_inf } | |
end open_subgroup | |
namespace submodule | |
open open_add_subgroup | |
variables {R : Type*} {M : Type*} [comm_ring R] | |
variables [add_comm_group M] [topological_space M] [topological_add_group M] [module R M] | |
lemma is_open_mono {U P : submodule R M} (h : U β€ P) (hU : is_open (U : set M)) : | |
is_open (P : set M) := | |
@add_subgroup.is_open_mono M _ _ _ U.to_add_subgroup P.to_add_subgroup h hU | |
end submodule | |
namespace ideal | |
variables {R : Type*} [comm_ring R] | |
variables [topological_space R] [topological_ring R] | |
lemma is_open_of_open_subideal {U I : ideal R} (h : U β€ I) (hU : is_open (U : set R)) : | |
is_open (I : set R) := | |
submodule.is_open_mono h hU | |
end ideal | |