feat(topology/hom/open): Continuous open maps (#12406)

Define `continuous_open_map`, the type of continuous opens maps between two topological spaces, and `continuous_open_map_class`, its companion hom class.

src/topology/hom/open.lean

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+/-
+Copyright (c) 2022 Yaël Dillies. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yaël Dillies
+-/
+import topology.continuous_function.basic
+
+/-!
+# Continuous open maps
+
+This file defines bundled continuous open 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.
+
+## Types of morphisms
+
+* `continuous_open_map`: Continuous open maps.
+
+## Typeclasses
+
+* `continuous_open_map_class`
+-/
+
+open function
+
+variables {F α β γ δ : Type*}
+
+/-- The type of continuous open maps from `α` to `β`, aka Priestley homomorphisms. -/
+structure continuous_open_map (α β : Type*) [topological_space α] [topological_space β]
+  extends continuous_map α β :=
+(map_open' : is_open_map to_fun)
+
+infixr ` →CO `:25 := continuous_open_map
+
+/-- `continuous_open_map_class F α β` states that `F` is a type of continuous open maps.
+
+You should extend this class when you extend `continuous_open_map`. -/
+class continuous_open_map_class (F : Type*) (α β : out_param $ Type*) [topological_space α]
+  [topological_space β] extends continuous_map_class F α β :=
+(map_open (f : F) : is_open_map f)
+
+export continuous_open_map_class (map_open)
+
+instance [topological_space α] [topological_space β] [continuous_open_map_class F α β] :
+  has_coe_t F (α →CO β) :=
+⟨λ f, ⟨f, map_open f⟩⟩
+
+/-! ### Continuous open maps -/
+
+namespace continuous_open_map
+variables [topological_space α] [topological_space β] [topological_space γ] [topological_space δ]
+
+instance : continuous_open_map_class (α →CO β) α β :=
+{ coe := λ f, f.to_fun,
+  coe_injective' := λ f g h, by { obtain ⟨⟨_, _⟩, _⟩ := f, obtain ⟨⟨_, _⟩, _⟩ := g, congr' },
+  map_continuous := λ f, f.continuous_to_fun,
+  map_open := λ f, f.map_open' }
+
+/-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun`
+directly. -/
+instance : has_coe_to_fun (α →CO β) (λ _, α → β) := fun_like.has_coe_to_fun
+
+@[simp] lemma to_fun_eq_coe {f : α →CO β} : f.to_fun = (f : α → β) := rfl
+
+@[ext] lemma ext {f g : α →CO β} (h : ∀ a, f a = g a) : f = g := fun_like.ext f g h
+
+/-- Copy of a `continuous_open_map` with a new `continuous_map` equal to the old one. Useful to fix
+definitional equalities. -/
+protected def copy (f : α →CO β) (f' : α → β) (h : f' = f) : α →CO β :=
+⟨f.to_continuous_map.copy f' $ by exact h, h.symm.subst f.map_open'⟩
+
+variables (α)
+
+/-- `id` as a `continuous_open_map`. -/
+protected def id : α →CO α := ⟨continuous_map.id _, is_open_map.id⟩
+
+instance : inhabited (α →CO α) := ⟨continuous_open_map.id _⟩
+
+@[simp] lemma coe_id : ⇑(continuous_open_map.id α) = id := rfl
+
+variables {α}
+
+@[simp] lemma id_apply (a : α) : continuous_open_map.id α a = a := rfl
+
+/-- Composition of `continuous_open_map`s as a `continuous_open_map`. -/
+def comp (f : β →CO γ) (g : α →CO β) : continuous_open_map α γ :=
+⟨f.to_continuous_map.comp g.to_continuous_map, f.map_open'.comp g.map_open'⟩
+
+@[simp] lemma coe_comp (f : β →CO γ) (g : α →CO β) : (f.comp g : α → γ) = f ∘ g := rfl
+@[simp] lemma comp_apply (f : β →CO γ) (g : α →CO β) (a : α) : (f.comp g) a = f (g a) := rfl
+@[simp] lemma comp_assoc (f : γ →CO δ) (g : β →CO γ) (h : α →CO β) :
+  (f.comp g).comp h = f.comp (g.comp h) := rfl
+@[simp] lemma comp_id (f : α →CO β) : f.comp (continuous_open_map.id α) = f := ext $ λ a, rfl
+@[simp] lemma id_comp (f : α →CO β) : (continuous_open_map.id β).comp f = f := ext $ λ a, rfl
+
+lemma cancel_right {g₁ g₂ : β →CO γ} {f : α →CO β} (hf : surjective f) :
+  g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=
+⟨λ h, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩
+
+lemma cancel_left {g : β →CO γ} {f₁ f₂ : α →CO β} (hg : injective g) :
+  g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
+⟨λ h, ext $ λ a, hg $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩
+
+end continuous_open_map