mathlib documentation

topology.discrete_quotient

Discrete quotients of a topological space. #

This file defines the type of discrete quotients of a topological space, denoted discrete_quotient X. To avoid quantifying over types, we model such quotients as setoids whose equivalence classes are clopen.

Definitions #

discrete_quotient X is the type of discrete quotients of X. It is endowed with a coercion to Type, which is defined as the quotient associated to the setoid in question, and each such quotient is endowed with the discrete topology.
Given S : discrete_quotient X, the projection X → S is denoted S.proj.
When X is compact and S : discrete_quotient X, the space S is endowed with a fintype instance.

Order structure #

The type discrete_quotient X is endowed with an instance of a semilattice_inf with order_top. The partial ordering A ≤ B mathematically means that B.proj factors through A.proj. The top element ⊤ is the trivial quotient, meaning that every element of X is collapsed to a point. Given h : A ≤ B, the map A → B is discrete_quotient.of_le h. Whenever X is discrete, the type discrete_quotient X is also endowed with an instance of a semilattice_inf with order_bot, where the bot element ⊥ is X itself.

Given f : X → Y and h : continuous f, we define a predicate le_comap h A B for A : discrete_quotient X and B : discrete_quotient Y, asserting that f descends to A → B. If cond : le_comap h A B, the function A → B is obtained by discrete_quotient.map cond.

Theorems #

The two main results proved in this file are:

discrete_quotient.eq_of_proj_eq which states that when X is compact, t2 and totally disconnected, any two elements of X agree if their projections in Q agree for all Q : discrete_quotient X.
discrete_quotient.exists_of_compat which states that when X is compact, then any system of elements of Q as Q : discrete_quotient X varies, which is compatible with respect to discrete_quotient.of_le, must arise from some element of X.

Remarks #

The constructions in this file will be used to show that any profinite space is a limit of finite discrete spaces.

@[ext]

structure discrete_quotient (X : Type u_1) [topological_space X] :

Type u_1

rel : X → X → Prop
equiv : equivalence self.rel
clopen : ∀ (x : X), is_clopen (set_of (self.rel x))

The type of discrete quotients of a topological space.

Instances for discrete_quotient

theorem discrete_quotient.ext_iff {X : Type u_1} {_inst_1 : topological_space X} (x y : discrete_quotient X) :

x = y ↔ x.rel = y.rel

theorem discrete_quotient.ext {X : Type u_1} {_inst_1 : topological_space X} (x y : discrete_quotient X) (h : x.rel = y.rel) :

x = y

def discrete_quotient.of_clopen {X : Type u_1} [topological_space X] {A : set X} (h : is_clopen A) :

discrete_quotient X

Construct a discrete quotient from a clopen set.

Equations

discrete_quotient.of_clopen h = {rel := λ (x y : X), x ∈ A ∧ y ∈ A ∨ x ∉ A ∧ y ∉ A, equiv := _, clopen := _}

theorem discrete_quotient.refl {X : Type u_1} [topological_space X] (S : discrete_quotient X) (x : X) :

S.rel x x

theorem discrete_quotient.symm {X : Type u_1} [topological_space X] (S : discrete_quotient X) (x y : X) :

S.rel x y → S.rel y x

theorem discrete_quotient.trans {X : Type u_1} [topological_space X] (S : discrete_quotient X) (x y z : X) :

S.rel x y → S.rel y z → S.rel x z

def discrete_quotient.setoid {X : Type u_1} [topological_space X] (S : discrete_quotient X) :

The setoid whose quotient yields the discrete quotient.

Equations

S.setoid = {r := S.rel, iseqv := _}

@[protected, instance]

def discrete_quotient.has_coe_to_sort {X : Type u_1} [topological_space X] :

has_coe_to_sort (discrete_quotient X) (Type u_1)

Equations

discrete_quotient.has_coe_to_sort = {coe := λ (S : discrete_quotient X), quotient S.setoid}

@[protected, instance]

def discrete_quotient.topological_space {X : Type u_1} [topological_space X] (S : discrete_quotient X) :

topological_space ↥S

Equations

S.topological_space = ⊥

def discrete_quotient.proj {X : Type u_1} [topological_space X] (S : discrete_quotient X) :

X → ↥S

The projection from X to the given discrete quotient.

Equations

S.proj = quotient.mk'

theorem discrete_quotient.proj_surjective {X : Type u_1} [topological_space X] (S : discrete_quotient X) :

function.surjective S.proj

theorem discrete_quotient.fiber_eq {X : Type u_1} [topological_space X] (S : discrete_quotient X) (x : X) :

S.proj ⁻¹' {S.proj x} = set_of (S.rel x)

theorem discrete_quotient.proj_is_locally_constant {X : Type u_1} [topological_space X] (S : discrete_quotient X) :

is_locally_constant S.proj

theorem discrete_quotient.proj_continuous {X : Type u_1} [topological_space X] (S : discrete_quotient X) :

continuous S.proj

theorem discrete_quotient.fiber_closed {X : Type u_1} [topological_space X] (S : discrete_quotient X) (A : set ↥S) :

is_closed (S.proj ⁻¹' A)

theorem discrete_quotient.fiber_open {X : Type u_1} [topological_space X] (S : discrete_quotient X) (A : set ↥S) :

is_open (S.proj ⁻¹' A)

theorem discrete_quotient.fiber_clopen {X : Type u_1} [topological_space X] (S : discrete_quotient X) (A : set ↥S) :

is_clopen (S.proj ⁻¹' A)

@[protected, instance]

def discrete_quotient.partial_order {X : Type u_1} [topological_space X] :

partial_order (discrete_quotient X)

Equations

discrete_quotient.partial_order = {le := λ (A B : discrete_quotient X), ∀ (x y : X), A.rel x y → B.rel x y, lt := preorder.lt._default (λ (A B : discrete_quotient X), ∀ (x y : X), A.rel x y → B.rel x y), le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _}

@[protected, instance]

def discrete_quotient.order_top {X : Type u_1} [topological_space X] :

order_top (discrete_quotient X)

Equations

discrete_quotient.order_top = {top := {rel := λ (a b : X), true, equiv := _, clopen := _}, le_top := _}

@[protected, instance]

def discrete_quotient.semilattice_inf {X : Type u_1} [topological_space X] :

semilattice_inf (discrete_quotient X)

Equations

discrete_quotient.semilattice_inf = {inf := λ (A B : discrete_quotient X), {rel := λ (x y : X), A.rel x y ∧ B.rel x y, equiv := _, clopen := _}, le := partial_order.le discrete_quotient.partial_order, lt := partial_order.lt discrete_quotient.partial_order, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, inf_le_left := _, inf_le_right := _, le_inf := _}

@[protected, instance]

def discrete_quotient.inhabited {X : Type u_1} [topological_space X] :

inhabited (discrete_quotient X)

Equations

discrete_quotient.inhabited = {default := ⊤}

def discrete_quotient.comap {X : Type u_1} [topological_space X] (S : discrete_quotient X) {Y : Type u_2} [topological_space Y] {f : Y → X} (cont : continuous f) :

discrete_quotient Y

Comap a discrete quotient along a continuous map.

Equations

S.comap cont = {rel := λ (a b : Y), S.rel (f a) (f b), equiv := _, clopen := _}

@[simp]

theorem discrete_quotient.comap_id {X : Type u_1} [topological_space X] (S : discrete_quotient X) :

S.comap continuous_id = S

@[simp]

theorem discrete_quotient.comap_comp {X : Type u_1} [topological_space X] (S : discrete_quotient X) {Y : Type u_2} [topological_space Y] {f : Y → X} (cont : continuous f) {Z : Type u_3} [topological_space Z] {g : Z → Y} (cont' : continuous g) :

S.comap _ = (S.comap cont).comap cont'

theorem discrete_quotient.comap_mono {X : Type u_1} [topological_space X] {Y : Type u_2} [topological_space Y] {f : Y → X} (cont : continuous f) {A B : discrete_quotient X} (h : A ≤ B) :

A.comap cont ≤ B.comap cont

def discrete_quotient.of_le {X : Type u_1} [topological_space X] {A B : discrete_quotient X} (h : A ≤ B) :

↥A → ↥B

The map induced by a refinement of a discrete quotient.

Equations

discrete_quotient.of_le h = λ (a : ↥A), quotient.lift_on' a (λ (x : X), B.proj x) _

@[simp]

theorem discrete_quotient.of_le_refl {X : Type u_1} [topological_space X] {A : discrete_quotient X} :

discrete_quotient.of_le _ = id

theorem discrete_quotient.of_le_refl_apply {X : Type u_1} [topological_space X] {A : discrete_quotient X} (a : ↥A) :

discrete_quotient.of_le _ a = a

@[simp]

theorem discrete_quotient.of_le_comp {X : Type u_1} [topological_space X] {A B C : discrete_quotient X} (h1 : A ≤ B) (h2 : B ≤ C) :

discrete_quotient.of_le _ = discrete_quotient.of_le h2 ∘ discrete_quotient.of_le h1

theorem discrete_quotient.of_le_comp_apply {X : Type u_1} [topological_space X] {A B C : discrete_quotient X} (h1 : A ≤ B) (h2 : B ≤ C) (a : ↥A) :

discrete_quotient.of_le _ a = discrete_quotient.of_le h2 (discrete_quotient.of_le h1 a)

theorem discrete_quotient.of_le_continuous {X : Type u_1} [topological_space X] {A B : discrete_quotient X} (h : A ≤ B) :

continuous (discrete_quotient.of_le h)

@[simp]

theorem discrete_quotient.of_le_proj {X : Type u_1} [topological_space X] {A B : discrete_quotient X} (h : A ≤ B) :

discrete_quotient.of_le h ∘ A.proj = B.proj

@[simp]

theorem discrete_quotient.of_le_proj_apply {X : Type u_1} [topological_space X] {A B : discrete_quotient X} (h : A ≤ B) (x : X) :

discrete_quotient.of_le h (A.proj x) = B.proj x

@[protected, instance]

def discrete_quotient.order_bot {X : Type u_1} [topological_space X] [discrete_topology X] :

order_bot (discrete_quotient X)

When X is discrete, there is a order_bot instance on discrete_quotient X

Equations

discrete_quotient.order_bot = {bot := {rel := eq X, equiv := _, clopen := _}, bot_le := _}

theorem discrete_quotient.proj_bot_injective {X : Type u_1} [topological_space X] [discrete_topology X] :

function.injective ⊥.proj

theorem discrete_quotient.proj_bot_bijective {X : Type u_1} [topological_space X] [discrete_topology X] :

function.bijective ⊥.proj

def discrete_quotient.le_comap {X : Type u_1} [topological_space X] {Y : Type u_2} [topological_space Y] {f : Y → X} (cont : continuous f) (A : discrete_quotient Y) (B : discrete_quotient X) :

Prop

Given cont : continuous f, le_comap cont A B is defined as A ≤ B.comap f. Mathematically this means that f descends to a morphism A → B.

Equations

discrete_quotient.le_comap cont A B = (A ≤ B.comap cont)

theorem discrete_quotient.le_comap_id {X : Type u_1} [topological_space X] (A : discrete_quotient X) :

discrete_quotient.le_comap continuous_id A A

theorem discrete_quotient.le_comap_comp {X : Type u_1} [topological_space X] {Y : Type u_2} [topological_space Y] {f : Y → X} {cont : continuous f} {A : discrete_quotient Y} {B : discrete_quotient X} {Z : Type u_3} [topological_space Z] {g : Z → Y} {cont' : continuous g} {C : discrete_quotient Z} :

discrete_quotient.le_comap cont' C A → discrete_quotient.le_comap cont A B → discrete_quotient.le_comap _ C B

theorem discrete_quotient.le_comap_trans {X : Type u_1} [topological_space X] {Y : Type u_2} [topological_space Y] {f : Y → X} {cont : continuous f} {A : discrete_quotient Y} {B C : discrete_quotient X} :

discrete_quotient.le_comap cont A B → B ≤ C → discrete_quotient.le_comap cont A C

def discrete_quotient.map {X : Type u_1} [topological_space X] {Y : Type u_2} [topological_space Y] {f : Y → X} {cont : continuous f} {A : discrete_quotient Y} {B : discrete_quotient X} (cond : discrete_quotient.le_comap cont A B) :

↥A → ↥B

Map a discrete quotient along a continuous map.

Equations

discrete_quotient.map cond = quotient.map' f cond

theorem discrete_quotient.map_continuous {X : Type u_1} [topological_space X] {Y : Type u_2} [topological_space Y] {f : Y → X} {cont : continuous f} {A : discrete_quotient Y} {B : discrete_quotient X} (cond : discrete_quotient.le_comap cont A B) :

continuous (discrete_quotient.map cond)

@[simp]

theorem discrete_quotient.map_proj {X : Type u_1} [topological_space X] {Y : Type u_2} [topological_space Y] {f : Y → X} {cont : continuous f} {A : discrete_quotient Y} {B : discrete_quotient X} (cond : discrete_quotient.le_comap cont A B) :

discrete_quotient.map cond ∘ A.proj = B.proj ∘ f

@[simp]

theorem discrete_quotient.map_proj_apply {X : Type u_1} [topological_space X] {Y : Type u_2} [topological_space Y] {f : Y → X} {cont : continuous f} {A : discrete_quotient Y} {B : discrete_quotient X} (cond : discrete_quotient.le_comap cont A B) (y : Y) :

discrete_quotient.map cond (A.proj y) = B.proj (f y)

@[simp]

theorem discrete_quotient.map_id {Y : Type u_2} [topological_space Y] {A : discrete_quotient Y} :

discrete_quotient.map _ = id

@[simp]

theorem discrete_quotient.map_comp {X : Type u_1} [topological_space X] {Y : Type u_2} [topological_space Y] {f : Y → X} {cont : continuous f} {A : discrete_quotient Y} {B : discrete_quotient X} {Z : Type u_3} [topological_space Z] {g : Z → Y} {cont' : continuous g} {C : discrete_quotient Z} (h1 : discrete_quotient.le_comap cont' C A) (h2 : discrete_quotient.le_comap cont A B) :

discrete_quotient.map _ = discrete_quotient.map h2 ∘ discrete_quotient.map h1

@[simp]

theorem discrete_quotient.of_le_map {X : Type u_1} [topological_space X] {Y : Type u_2} [topological_space Y] {f : Y → X} {cont : continuous f} {A : discrete_quotient Y} {B C : discrete_quotient X} (cond : discrete_quotient.le_comap cont A B) (h : B ≤ C) :

discrete_quotient.map _ = discrete_quotient.of_le h ∘ discrete_quotient.map cond

@[simp]

theorem discrete_quotient.of_le_map_apply {X : Type u_1} [topological_space X] {Y : Type u_2} [topological_space Y] {f : Y → X} {cont : continuous f} {A : discrete_quotient Y} {B C : discrete_quotient X} (cond : discrete_quotient.le_comap cont A B) (h : B ≤ C) (a : ↥A) :

discrete_quotient.map _ a = discrete_quotient.of_le h (discrete_quotient.map cond a)

@[simp]

theorem discrete_quotient.map_of_le {X : Type u_1} [topological_space X] {Y : Type u_2} [topological_space Y] {f : Y → X} {cont : continuous f} {A : discrete_quotient Y} {B : discrete_quotient X} {C : discrete_quotient Y} (cond : discrete_quotient.le_comap cont A B) (h : C ≤ A) :

discrete_quotient.map _ = discrete_quotient.map cond ∘ discrete_quotient.of_le h

@[simp]

theorem discrete_quotient.map_of_le_apply {X : Type u_1} [topological_space X] {Y : Type u_2} [topological_space Y] {f : Y → X} {cont : continuous f} {A : discrete_quotient Y} {B : discrete_quotient X} {C : discrete_quotient Y} (cond : discrete_quotient.le_comap cont A B) (h : C ≤ A) (c : ↥C) :

discrete_quotient.map _ c = discrete_quotient.map cond (discrete_quotient.of_le h c)

theorem discrete_quotient.eq_of_proj_eq {X : Type u_1} [topological_space X] [t2_space X] [compact_space X] [disc : totally_disconnected_space X] {x y : X} :

(∀ (Q : discrete_quotient X), Q.proj x = Q.proj y) → x = y

theorem discrete_quotient.fiber_le_of_le {X : Type u_1} [topological_space X] {A B : discrete_quotient X} (h : A ≤ B) (a : ↥A) :

A.proj ⁻¹' {a} ≤ B.proj ⁻¹' {discrete_quotient.of_le h a}

theorem discrete_quotient.exists_of_compat {X : Type u_1} [topological_space X] [compact_space X] (Qs : Π (Q : discrete_quotient X), ↥Q) (compat : ∀ (A B : discrete_quotient X) (h : A ≤ B), discrete_quotient.of_le h (Qs A) = Qs B) :

∃ (x : X), ∀ (Q : discrete_quotient X), Q.proj x = Qs Q

@[protected, instance]

noncomputable def discrete_quotient.fintype {X : Type u_1} [topological_space X] (S : discrete_quotient X) [compact_space X] :

Equations

S.fintype = let T : finset ↥S := classical.some _ in {elems := T, complete := _}

def locally_constant.discrete_quotient {X : Type u_1} [topological_space X] {α : Type u_2} (f : locally_constant X α) :

discrete_quotient X

Any locally constant function induces a discrete quotient.

Equations

f.discrete_quotient = {rel := λ (a b : X), ⇑f b = ⇑f a, equiv := _, clopen := _}

def locally_constant.lift {X : Type u_1} [topological_space X] {α : Type u_2} (f : locally_constant X α) :

↥(f.discrete_quotient) → α

The function from the discrete quotient associated to a locally constant function.

Equations

f.lift = λ (a : ↥(f.discrete_quotient)), quotient.lift_on' a ⇑f _

theorem locally_constant.lift_is_locally_constant {X : Type u_1} [topological_space X] {α : Type u_2} (f : locally_constant X α) :

is_locally_constant f.lift

def locally_constant.locally_constant_lift {X : Type u_1} [topological_space X] {α : Type u_2} (f : locally_constant X α) :

locally_constant ↥(f.discrete_quotient) α

A locally constant version of locally_constant.lift.

Equations

f.locally_constant_lift = {to_fun := f.lift, is_locally_constant := _}

@[simp]

theorem locally_constant.lift_eq_coe {X : Type u_1} [topological_space X] {α : Type u_2} (f : locally_constant X α) :

f.lift = ⇑(f.locally_constant_lift)

@[simp]

theorem locally_constant.factors {X : Type u_1} [topological_space X] {α : Type u_2} (f : locally_constant X α) :

⇑(f.locally_constant_lift) ∘ f.discrete_quotient.proj = ⇑f