Continuous open maps

This file defines bundled continuous open maps.

We use the DFunLike 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

• ContinuousOpenMap: Continuous open maps.

Typeclasses

• ContinuousOpenMapClass

structure ContinuousOpenMap (α : Type u_6) (β : Type u_7) [TopologicalSpace α] [TopologicalSpace β] extends ContinuousMap : Type (max u_6 u_7)

The type of continuous open maps from α to β, aka Priestley homomorphisms.

• toFun : α → β
• continuous_toFun : Continuous self.toFun
• map_open' : IsOpenMap self.toFun

def «term_→CO_» : Lean.TrailingParserDescr

Equations

• «term_→CO_» = Lean.ParserDescr.trailingNode `term_→CO_ 25 26 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " →CO ") (Lean.ParserDescr.cat `term 25))

class ContinuousOpenMapClass (F : Type u_6) (α : outParam (Type u_7)) (β : outParam (Type u_8)) [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] extends ContinuousMapClass : Prop

ContinuousOpenMapClass F α β states that F is a type of continuous open maps.

You should extend this class when you extend ContinuousOpenMap.

• map_continuous : ∀ (f : F), Continuous ⇑f
• map_open : ∀ (f : F), IsOpenMap ⇑f

instance instCoeTCContinuousOpenMap {F : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] [ContinuousOpenMapClass F α β] : CoeTC F (α →CO β)

Equations

• instCoeTCContinuousOpenMap = { coe := fun (f : F) => { toContinuousMap := ↑f, map_open' := (_ : IsOpenMap ⇑f) } }

Continuous open maps

instance ContinuousOpenMap.instFunLike {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] : FunLike (α →CO β) α β

Equations

• One or more equations did not get rendered due to their size.

instance ContinuousOpenMap.instContinuousOpenMapClassContinuousOpenMapInstFunLike {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] : ContinuousOpenMapClass (α →CO β) α β

Equations

• (_ : ContinuousOpenMapClass (α →CO β) α β) = (_ : ContinuousOpenMapClass (α →CO β) α β)

theorem ContinuousOpenMap.toFun_eq_coe {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {f : α →CO β} : f.toFun = ⇑f

@[simp]

theorem ContinuousOpenMap.coe_toContinuousMap {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (f : α →CO β) : ⇑f.toContinuousMap = ⇑f

theorem ContinuousOpenMap.ext {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {f : α →CO β} {g : α →CO β} (h : ∀ (a : α), f a = g a) : f = g

def ContinuousOpenMap.copy {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (f : α →CO β) (f' : α → β) (h : f' = ⇑f) : α →CO β

Copy of a ContinuousOpenMap with a new ContinuousMap equal to the old one. Useful to fix definitional equalities.

Equations

• ContinuousOpenMap.copy f f' h = { toContinuousMap := ContinuousMap.copy f.toContinuousMap f' h, map_open' := (_ : IsOpenMap f') }

@[simp]

theorem ContinuousOpenMap.coe_copy {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (f : α →CO β) (f' : α → β) (h : f' = ⇑f) : ⇑(ContinuousOpenMap.copy f f' h) = f'

theorem ContinuousOpenMap.copy_eq {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (f : α →CO β) (f' : α → β) (h : f' = ⇑f) : ContinuousOpenMap.copy f f' h = f

def ContinuousOpenMap.id (α : Type u_2) [TopologicalSpace α] : α →CO α

id as a ContinuousOpenMap.

Equations

• ContinuousOpenMap.id α = { toContinuousMap := ContinuousMap.id α, map_open' := (_ : IsOpenMap id) }

instance ContinuousOpenMap.instInhabitedContinuousOpenMap (α : Type u_2) [TopologicalSpace α] : Inhabited (α →CO α)

Equations

• ContinuousOpenMap.instInhabitedContinuousOpenMap α = { default := ContinuousOpenMap.id α }

@[simp]

theorem ContinuousOpenMap.coe_id (α : Type u_2) [TopologicalSpace α] : ⇑(ContinuousOpenMap.id α) = id

@[simp]

theorem ContinuousOpenMap.id_apply {α : Type u_2} [TopologicalSpace α] (a : α) : (ContinuousOpenMap.id α) a = a

def ContinuousOpenMap.comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : β →CO γ) (g : α →CO β) : α →CO γ

Composition of ContinuousOpenMaps as a ContinuousOpenMap.

Equations

• ContinuousOpenMap.comp f g = { toContinuousMap := ContinuousMap.comp f.toContinuousMap g.toContinuousMap, map_open' := (_ : IsOpenMap (f.toFun ∘ fun (x : α) => g.toContinuousMap x)) }

@[simp]

theorem ContinuousOpenMap.coe_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : β →CO γ) (g : α →CO β) : ⇑(ContinuousOpenMap.comp f g) = ⇑f ∘ ⇑g

@[simp]

theorem ContinuousOpenMap.comp_apply {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : β →CO γ) (g : α →CO β) (a : α) : (ContinuousOpenMap.comp f g) a = f (g a)

@[simp]

theorem ContinuousOpenMap.comp_assoc {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (f : γ →CO δ) (g : β →CO γ) (h : α →CO β) : ContinuousOpenMap.comp (ContinuousOpenMap.comp f g) h = ContinuousOpenMap.comp f (ContinuousOpenMap.comp g h)

@[simp]

theorem ContinuousOpenMap.comp_id {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (f : α →CO β) : ContinuousOpenMap.comp f (ContinuousOpenMap.id α) = f

@[simp]

theorem ContinuousOpenMap.id_comp {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (f : α →CO β) : ContinuousOpenMap.comp (ContinuousOpenMap.id β) f = f

@[simp]

theorem ContinuousOpenMap.cancel_right {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {g₁ : β →CO γ} {g₂ : β →CO γ} {f : α →CO β} (hf : Function.Surjective ⇑f) : ContinuousOpenMap.comp g₁ f = ContinuousOpenMap.comp g₂ f ↔ g₁ = g₂

@[simp]

theorem ContinuousOpenMap.cancel_left {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {g : β →CO γ} {f₁ : α →CO β} {f₂ : α →CO β} (hg : Function.Injective ⇑g) : ContinuousOpenMap.comp g f₁ = ContinuousOpenMap.comp g f₂ ↔ f₁ = f₂