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/-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri
-/
import data.set.Union_lift
import topology.homeomorph
/-!
# Continuous bundled maps
In this file we define the type `continuous_map` of continuous bundled maps.
We use the `fun_like` design, so each type of morphisms has a companion typeclass which is meant to
be satisfied by itself and all stricter types.
-/
open function
/-- The type of continuous maps from `α` to `β`.
When possible, instead of parametrizing results over `(f : C(α, β))`,
you should parametrize over `{F : Type*} [continuous_map_class F α β] (f : F)`.
When you extend this structure, make sure to extend `continuous_map_class`. -/
@[protect_proj]
structure continuous_map (α β : Type*) [topological_space α] [topological_space β] :=
(to_fun : α → β)
(continuous_to_fun : continuous to_fun . tactic.interactive.continuity')
notation `C(` α `, ` β `)` := continuous_map α β
/-- `continuous_map_class F α β` states that `F` is a type of continuous maps.
You should extend this class when you extend `continuous_map`. -/
class continuous_map_class (F : Type*) (α β : out_param $ Type*) [topological_space α]
[topological_space β]
extends fun_like F α (λ _, β) :=
(map_continuous (f : F) : continuous f)
export continuous_map_class (map_continuous)
attribute [continuity] map_continuous
section continuous_map_class
variables {F α β : Type*} [topological_space α] [topological_space β] [continuous_map_class F α β]
include β
lemma map_continuous_at (f : F) (a : α) : continuous_at f a := (map_continuous f).continuous_at
lemma map_continuous_within_at (f : F) (s : set α) (a : α) : continuous_within_at f s a :=
(map_continuous f).continuous_within_at
instance : has_coe_t F C(α, β) := ⟨λ f, { to_fun := f, continuous_to_fun := map_continuous f }⟩
end continuous_map_class
/-! ### Continuous maps-/
namespace continuous_map
variables {α β γ δ : Type*} [topological_space α] [topological_space β] [topological_space γ]
[topological_space δ]
instance : continuous_map_class C(α, β) α β :=
{ coe := continuous_map.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
map_continuous := continuous_map.continuous_to_fun }
/-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun`
directly. -/
instance : has_coe_to_fun (C(α, β)) (λ _, α → β) := fun_like.has_coe_to_fun
@[simp] lemma to_fun_eq_coe {f : C(α, β)} : f.to_fun = (f : α → β) := rfl
-- this must come after the coe_to_fun definition
initialize_simps_projections continuous_map (to_fun → apply)
@[ext] lemma ext {f g : C(α, β)} (h : ∀ a, f a = g a) : f = g := fun_like.ext _ _ h
/-- Copy of a `continuous_map` with a new `to_fun` equal to the old one. Useful to fix definitional
equalities. -/
protected def copy (f : C(α, β)) (f' : α → β) (h : f' = f) : C(α, β) :=
{ to_fun := f',
continuous_to_fun := h.symm ▸ f.continuous_to_fun }
variables {α β} {f g : C(α, β)}
/-- Deprecated. Use `map_continuous` instead. -/
protected lemma continuous (f : C(α, β)) : continuous f := f.continuous_to_fun
@[continuity] lemma continuous_set_coe (s : set C(α, β)) (f : s) : continuous f := f.1.continuous
/-- Deprecated. Use `map_continuous_at` instead. -/
protected lemma continuous_at (f : C(α, β)) (x : α) : continuous_at f x :=
f.continuous.continuous_at
/-- Deprecated. Use `fun_like.congr_fun` instead. -/
protected lemma congr_fun {f g : C(α, β)} (H : f = g) (x : α) : f x = g x := H ▸ rfl
/-- Deprecated. Use `fun_like.congr_arg` instead. -/
protected lemma congr_arg (f : C(α, β)) {x y : α} (h : x = y) : f x = f y := h ▸ rfl
lemma coe_injective : @function.injective (C(α, β)) (α → β) coe_fn :=
λ f g h, by cases f; cases g; congr'
@[simp] lemma coe_mk (f : α → β) (h : continuous f) :
⇑(⟨f, h⟩ : C(α, β)) = f := rfl
lemma map_specializes (f : C(α, β)) {x y : α} (h : x ⤳ y) : f x ⤳ f y := h.map f.2
section
variables (α β)
/--
The continuous functions from `α` to `β` are the same as the plain functions when `α` is discrete.
-/
@[simps]
def equiv_fn_of_discrete [discrete_topology α] : C(α, β) ≃ (α → β) :=
⟨(λ f, f), (λ f, ⟨f, continuous_of_discrete_topology⟩),
λ f, by { ext, refl, }, λ f, by { ext, refl, }⟩
end
variables (α)
/-- The identity as a continuous map. -/
protected def id : C(α, α) := ⟨id⟩
@[simp] lemma coe_id : ⇑(continuous_map.id α) = id := rfl
/-- The constant map as a continuous map. -/
def const (b : β) : C(α, β) := ⟨const α b⟩
@[simp] lemma coe_const (b : β) : ⇑(const α b) = function.const α b := rfl
instance [inhabited β] : inhabited C(α, β) :=
⟨const α default⟩
variables {α}
@[simp] lemma id_apply (a : α) : continuous_map.id α a = a := rfl
@[simp] lemma const_apply (b : β) (a : α) : const α b a = b := rfl
/-- The composition of continuous maps, as a continuous map. -/
def comp (f : C(β, γ)) (g : C(α, β)) : C(α, γ) := ⟨f ∘ g⟩
@[simp] lemma coe_comp (f : C(β, γ)) (g : C(α, β)) : ⇑(comp f g) = f ∘ g := rfl
@[simp] lemma comp_apply (f : C(β, γ)) (g : C(α, β)) (a : α) : comp f g a = f (g a) := rfl
@[simp] lemma comp_assoc (f : C(γ, δ)) (g : C(β, γ)) (h : C(α, β)) :
(f.comp g).comp h = f.comp (g.comp h) := rfl
@[simp] lemma id_comp (f : C(α, β)) : (continuous_map.id _).comp f = f := ext $ λ _, rfl
@[simp] lemma comp_id (f : C(α, β)) : f.comp (continuous_map.id _) = f := ext $ λ _, rfl
@[simp] lemma const_comp (c : γ) (f : C(α, β)) : (const β c).comp f = const α c := ext $ λ _, rfl
@[simp] lemma comp_const (f : C(β, γ)) (b : β) : f.comp (const α b) = const α (f b) :=
ext $ λ _, rfl
lemma cancel_right {f₁ f₂ : C(β, γ)} {g : C(α, β)} (hg : surjective g) :
f₁.comp g = f₂.comp g ↔ f₁ = f₂ :=
⟨λ h, ext $ hg.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩
lemma cancel_left {f : C(β, γ)} {g₁ g₂ : C(α, β)} (hf : injective f) :
f.comp g₁ = f.comp g₂ ↔ g₁ = g₂ :=
⟨λ h, ext $ λ a, hf $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩
instance [nonempty α] [nontrivial β] : nontrivial C(α, β) :=
⟨let ⟨b₁, b₂, hb⟩ := exists_pair_ne β in
⟨const _ b₁, const _ b₂, λ h, hb $ fun_like.congr_fun h $ classical.arbitrary α⟩⟩
section prod
variables {α₁ α₂ β₁ β₂ : Type*}
[topological_space α₁] [topological_space α₂]
[topological_space β₁] [topological_space β₂]
/-- Given two continuous maps `f` and `g`, this is the continuous map `x ↦ (f x, g x)`. -/
def prod_mk (f : C(α, β₁)) (g : C(α, β₂)) :
C(α, β₁ × β₂) :=
{ to_fun := (λ x, (f x, g x)),
continuous_to_fun := continuous.prod_mk f.continuous g.continuous }
/-- Given two continuous maps `f` and `g`, this is the continuous map `(x, y) ↦ (f x, g y)`. -/
@[simps] def prod_map (f : C(α₁, α₂)) (g : C(β₁, β₂)) :
C(α₁ × β₁, α₂ × β₂) :=
{ to_fun := prod.map f g,
continuous_to_fun := continuous.prod_map f.continuous g.continuous }
@[simp] lemma prod_eval (f : C(α, β₁)) (g : C(α, β₂)) (a : α) :
(prod_mk f g) a = (f a, g a) := rfl
end prod
section pi
variables {I A : Type*} {X : I → Type*}
[topological_space A] [∀ i, topological_space (X i)]
/-- Abbreviation for product of continuous maps, which is continuous -/
def pi (f : Π i, C(A, X i)) : C(A, Π i, X i) :=
{ to_fun := λ (a : A) (i : I), f i a, }
@[simp] lemma pi_eval (f : Π i, C(A, X i)) (a : A) :
(pi f) a = λ i : I, (f i) a := rfl
end pi
section restrict
variables (s : set α)
/-- The restriction of a continuous function `α → β` to a subset `s` of `α`. -/
def restrict (f : C(α, β)) : C(s, β) := ⟨f ∘ coe⟩
@[simp] lemma coe_restrict (f : C(α, β)) : ⇑(f.restrict s) = f ∘ coe := rfl
end restrict
section gluing
variables {ι : Type*}
(S : ι → set α)
(φ : Π i : ι, C(S i, β))
(hφ : ∀ i j (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), φ i ⟨x, hxi⟩ = φ j ⟨x, hxj⟩)
(hS : ∀ x : α, ∃ i, S i ∈ nhds x)
include hφ hS
/-- A family `φ i` of continuous maps `C(S i, β)`, where the domains `S i` contain a neighbourhood
of each point in `α` and the functions `φ i` agree pairwise on intersections, can be glued to
construct a continuous map in `C(α, β)`. -/
noncomputable def lift_cover : C(α, β) :=
begin
have H : (⋃ i, S i) = set.univ,
{ rw set.eq_univ_iff_forall,
intros x,
rw set.mem_Union,
obtain ⟨i, hi⟩ := hS x,
exact ⟨i, mem_of_mem_nhds hi⟩ },
refine ⟨set.lift_cover S (λ i, φ i) hφ H, continuous_subtype_nhds_cover hS _⟩,
intros i,
convert (φ i).continuous,
ext x,
exact set.lift_cover_coe x,
end
variables {S φ hφ hS}
@[simp] lemma lift_cover_coe {i : ι} (x : S i) : lift_cover S φ hφ hS x = φ i x :=
set.lift_cover_coe _
@[simp] lemma lift_cover_restrict {i : ι} : (lift_cover S φ hφ hS).restrict (S i) = φ i :=
ext $ lift_cover_coe
omit hφ hS
variables (A : set (set α))
(F : Π (s : set α) (hi : s ∈ A), C(s, β))
(hF : ∀ s (hs : s ∈ A) t (ht : t ∈ A) (x : α) (hxi : x ∈ s) (hxj : x ∈ t),
F s hs ⟨x, hxi⟩ = F t ht ⟨x, hxj⟩)
(hA : ∀ x : α, ∃ i ∈ A, i ∈ nhds x)
include hF hA
/-- A family `F s` of continuous maps `C(s, β)`, where (1) the domains `s` are taken from a set `A`
of sets in `α` which contain a neighbourhood of each point in `α` and (2) the functions `F s` agree
pairwise on intersections, can be glued to construct a continuous map in `C(α, β)`. -/
noncomputable def lift_cover' : C(α, β) :=
begin
let S : A → set α := coe,
let F : Π i : A, C(i, β) := λ i, F i i.prop,
refine lift_cover S F (λ i j, hF i i.prop j j.prop) _,
intros x,
obtain ⟨s, hs, hsx⟩ := hA x,
exact ⟨⟨s, hs⟩, hsx⟩
end
variables {A F hF hA}
@[simp] lemma lift_cover_coe' {s : set α} {hs : s ∈ A} (x : s) :
lift_cover' A F hF hA x = F s hs x :=
let x' : (coe : A → set α) ⟨s, hs⟩ := x in lift_cover_coe x'
@[simp] lemma lift_cover_restrict' {s : set α} {hs : s ∈ A} :
(lift_cover' A F hF hA).restrict s = F s hs :=
ext $ lift_cover_coe'
end gluing
end continuous_map
/--
The forward direction of a homeomorphism, as a bundled continuous map.
-/
@[simps]
def homeomorph.to_continuous_map {α β : Type*} [topological_space α] [topological_space β]
(e : α ≃ₜ β) : C(α, β) := ⟨e⟩
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