Continuous bundled maps

In this file we define the type ContinuousMap of continuous bundled 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.

structure ContinuousMap (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [TopologicalSpace β] : Type (max u_1 u_2)

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*} [ContinuousMapClass F α β] (f : F).

When you extend this structure, make sure to extend ContinuousMapClass.

• toFun : α → β

  The function α → β

• continuous_toFun : Continuous self.toFun

  Proposition that toFun is continuous

theorem ContinuousMap.continuous_toFun {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (self : C(α, β)) : Continuous self.toFun

Proposition that toFun is continuous

def «termC(_,_)» : Lean.ParserDescr

The type of continuous maps from α to β.

Equations

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

class ContinuousMapClass (F : Type u_1) (α : Type u_2) (β : Type u_3) [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] : Prop

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

You should extend this class when you extend ContinuousMap.

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

  Continuity

theorem ContinuousMapClass.map_continuous {F : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] [self : ContinuousMapClass F α β] (f : F) : Continuous ⇑f

Continuity

theorem map_continuousAt {F : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] [ContinuousMapClass F α β] (f : F) (a : α) : ContinuousAt (⇑f) a

theorem map_continuousWithinAt {F : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] [ContinuousMapClass F α β] (f : F) (s : Set α) (a : α) : ContinuousWithinAt (⇑f) s a

def toContinuousMap {F : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] [ContinuousMapClass F α β] (f : F) : C(α, β)

Coerce a bundled morphism with a ContinuousMapClass instance to a ContinuousMap.

Equations

• ↑f = { toFun := ⇑f, continuous_toFun := ⋯ }

instance instCoeTCContinuousMap {F : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] [ContinuousMapClass F α β] : CoeTC F C(α, β)

Equations

• instCoeTCContinuousMap = { coe := toContinuousMap }

Continuous maps

instance ContinuousMap.funLike {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : FunLike C(α, β) α β

Equations

• ContinuousMap.funLike = { coe := ContinuousMap.toFun, coe_injective' := ⋯ }

instance ContinuousMap.toContinuousMapClass {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : ContinuousMapClass C(α, β) α β

Equations

• ⋯ = ⋯

@[simp]

theorem ContinuousMap.toFun_eq_coe {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : C(α, β)} : f.toFun = ⇑f

instance ContinuousMap.instCanLiftForallCoeContinuous {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : CanLift (α → β) C(α, β) DFunLike.coe Continuous

Equations

• ⋯ = ⋯

def ContinuousMap.Simps.apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) : α → β

See note [custom simps projection].

Equations

• ContinuousMap.Simps.apply f = ⇑f

@[simp]

theorem ContinuousMap.coe_coe {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {F : Type u_5} [FunLike F α β] [ContinuousMapClass F α β] (f : F) : ⇑↑f = ⇑f

theorem ContinuousMap.ext {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : C(α, β)} {g : C(α, β)} (h : ∀ (a : α), f a = g a) : f = g

def ContinuousMap.copy {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (f' : α → β) (h : f' = ⇑f) : C(α, β)

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

Equations

• f.copy f' h = { toFun := f', continuous_toFun := ⋯ }

@[simp]

theorem ContinuousMap.coe_copy {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (f' : α → β) (h : f' = ⇑f) : ⇑(f.copy f' h) = f'

theorem ContinuousMap.copy_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (f' : α → β) (h : f' = ⇑f) : f.copy f' h = f

theorem ContinuousMap.continuous {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) : Continuous ⇑f

Deprecated. Use map_continuous instead.

theorem ContinuousMap.continuous_set_coe {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (s : Set C(α, β)) (f : ↑s) : Continuous ⇑↑f

theorem ContinuousMap.continuousAt {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (x : α) : ContinuousAt (⇑f) x

Deprecated. Use map_continuousAt instead.

theorem ContinuousMap.congr_fun {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : C(α, β)} {g : C(α, β)} (H : f = g) (x : α) : f x = g x

Deprecated. Use DFunLike.congr_fun instead.

theorem ContinuousMap.congr_arg {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) {x : α} {y : α} (h : x = y) : f x = f y

Deprecated. Use DFunLike.congr_arg instead.

theorem ContinuousMap.coe_injective {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : Function.Injective DFunLike.coe

@[simp]

theorem ContinuousMap.coe_mk {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) (h : Continuous f) : ⇑{ toFun := f, continuous_toFun := h } = f

theorem ContinuousMap.map_specializes {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) {x : α} {y : α} (h : x ⤳ y) : f x ⤳ f y

@[simp]

theorem ContinuousMap.equivFnOfDiscrete_symm_apply_apply (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [TopologicalSpace β] [DiscreteTopology α] (f : α → β) : ∀ (a : α), ((ContinuousMap.equivFnOfDiscrete α β).symm f) a = f a

@[simp]

theorem ContinuousMap.equivFnOfDiscrete_apply (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [TopologicalSpace β] [DiscreteTopology α] (f : C(α, β)) (a : α) : (ContinuousMap.equivFnOfDiscrete α β) f a = f a

def ContinuousMap.equivFnOfDiscrete (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [TopologicalSpace β] [DiscreteTopology α] : C(α, β) ≃ (α → β)

The continuous functions from α to β are the same as the plain functions when α is discrete.

Equations

• ContinuousMap.equivFnOfDiscrete α β = { toFun := fun (f : C(α, β)) => ⇑f, invFun := fun (f : α → β) => { toFun := f, continuous_toFun := ⋯ }, left_inv := ⋯, right_inv := ⋯ }

def ContinuousMap.id (α : Type u_1) [TopologicalSpace α] : C(α, α)

The identity as a continuous map.

Equations

• ContinuousMap.id α = { toFun := id, continuous_toFun := ⋯ }

@[simp]

theorem ContinuousMap.coe_id (α : Type u_1) [TopologicalSpace α] : ⇑(ContinuousMap.id α) = id

def ContinuousMap.const (α : Type u_1) {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (b : β) : C(α, β)

The constant map as a continuous map.

Equations

• ContinuousMap.const α b = { toFun := fun (x : α) => b, continuous_toFun := ⋯ }

@[simp]

theorem ContinuousMap.coe_const (α : Type u_1) {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (b : β) : ⇑(ContinuousMap.const α b) = Function.const α b

@[simp]

theorem ContinuousMap.constPi_apply (α : Type u_1) {β : Type u_2} [TopologicalSpace β] : ⇑(ContinuousMap.constPi α) = fun (b : β) => Function.const α b

def ContinuousMap.constPi (α : Type u_1) {β : Type u_2} [TopologicalSpace β] : C(β, α → β)

Function.const α b as a bundled continuous function of b.

Equations

• ContinuousMap.constPi α = { toFun := fun (b : β) => Function.const α b, continuous_toFun := ⋯ }

instance ContinuousMap.instInhabited (α : Type u_1) {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Inhabited β] : Inhabited C(α, β)

Equations

• ContinuousMap.instInhabited α = { default := ContinuousMap.const α default }

@[simp]

theorem ContinuousMap.id_apply {α : Type u_1} [TopologicalSpace α] (a : α) : (ContinuousMap.id α) a = a

@[simp]

theorem ContinuousMap.const_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (b : β) (a : α) : (ContinuousMap.const α b) a = b

def ContinuousMap.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : C(β, γ)) (g : C(α, β)) : C(α, γ)

The composition of continuous maps, as a continuous map.

Equations

• f.comp g = { toFun := ⇑f ∘ ⇑g, continuous_toFun := ⋯ }

@[simp]

theorem ContinuousMap.coe_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : C(β, γ)) (g : C(α, β)) : ⇑(f.comp g) = ⇑f ∘ ⇑g

@[simp]

theorem ContinuousMap.comp_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : C(β, γ)) (g : C(α, β)) (a : α) : (f.comp g) a = f (g a)

@[simp]

theorem ContinuousMap.comp_assoc {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (f : C(γ, δ)) (g : C(β, γ)) (h : C(α, β)) : (f.comp g).comp h = f.comp (g.comp h)

@[simp]

theorem ContinuousMap.id_comp {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) : (ContinuousMap.id β).comp f = f

@[simp]

theorem ContinuousMap.comp_id {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) : f.comp (ContinuousMap.id α) = f

@[simp]

theorem ContinuousMap.const_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (c : γ) (f : C(α, β)) : (ContinuousMap.const β c).comp f = ContinuousMap.const α c

@[simp]

theorem ContinuousMap.comp_const {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : C(β, γ)) (b : β) : f.comp (ContinuousMap.const α b) = ContinuousMap.const α (f b)

@[simp]

theorem ContinuousMap.cancel_right {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f₁ : C(β, γ)} {f₂ : C(β, γ)} {g : C(α, β)} (hg : Function.Surjective ⇑g) : f₁.comp g = f₂.comp g ↔ f₁ = f₂

@[simp]

theorem ContinuousMap.cancel_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : C(β, γ)} {g₁ : C(α, β)} {g₂ : C(α, β)} (hf : Function.Injective ⇑f) : f.comp g₁ = f.comp g₂ ↔ g₁ = g₂

instance ContinuousMap.instNontrivialOfNonempty {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Nonempty α] [Nontrivial β] : Nontrivial C(α, β)

Equations

• ⋯ = ⋯

@[simp]

theorem ContinuousMap.fst_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : ⇑ContinuousMap.fst = Prod.fst

def ContinuousMap.fst {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : C(α × β, α)

Prod.fst : (x, y) ↦ x as a bundled continuous map.

Equations

• ContinuousMap.fst = { toFun := Prod.fst, continuous_toFun := ⋯ }

@[simp]

theorem ContinuousMap.snd_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : ⇑ContinuousMap.snd = Prod.snd

def ContinuousMap.snd {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : C(α × β, β)

Prod.snd : (x, y) ↦ y as a bundled continuous map.

Equations

• ContinuousMap.snd = { toFun := Prod.snd, continuous_toFun := ⋯ }

def ContinuousMap.prodMk {α : Type u_1} [TopologicalSpace α] {β₁ : Type u_7} {β₂ : Type u_8} [TopologicalSpace β₁] [TopologicalSpace β₂] (f : C(α, β₁)) (g : C(α, β₂)) : C(α, β₁ × β₂)

Given two continuous maps f and g, this is the continuous map x ↦ (f x, g x).

Equations

• f.prodMk g = { toFun := fun (x : α) => (f x, g x), continuous_toFun := ⋯ }

@[simp]

theorem ContinuousMap.prodMap_apply {α₁ : Type u_5} {α₂ : Type u_6} {β₁ : Type u_7} {β₂ : Type u_8} [TopologicalSpace α₁] [TopologicalSpace α₂] [TopologicalSpace β₁] [TopologicalSpace β₂] (f : C(α₁, α₂)) (g : C(β₁, β₂)) : ∀ (a : α₁ × β₁), (f.prodMap g) a = Prod.map (⇑f) (⇑g) a

def ContinuousMap.prodMap {α₁ : Type u_5} {α₂ : Type u_6} {β₁ : Type u_7} {β₂ : Type u_8} [TopologicalSpace α₁] [TopologicalSpace α₂] [TopologicalSpace β₁] [TopologicalSpace β₂] (f : C(α₁, α₂)) (g : C(β₁, β₂)) : C(α₁ × β₁, α₂ × β₂)

Given two continuous maps f and g, this is the continuous map (x, y) ↦ (f x, g y).

Equations

• f.prodMap g = { toFun := Prod.map ⇑f ⇑g, continuous_toFun := ⋯ }

@[simp]

theorem ContinuousMap.prod_eval {α : Type u_1} [TopologicalSpace α] {β₁ : Type u_7} {β₂ : Type u_8} [TopologicalSpace β₁] [TopologicalSpace β₂] (f : C(α, β₁)) (g : C(α, β₂)) (a : α) : (f.prodMk g) a = (f a, g a)

@[simp]

theorem ContinuousMap.prodSwap_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (x : α × β) : ContinuousMap.prodSwap x = (x.2, x.1)

def ContinuousMap.prodSwap {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : C(α × β, β × α)

Prod.swap bundled as a ContinuousMap.

Equations

• ContinuousMap.prodSwap = ContinuousMap.snd.prodMk ContinuousMap.fst

@[simp]

theorem ContinuousMap.sigmaMk_apply {I : Type u_5} {X : I → Type u_7} [(i : I) → TopologicalSpace (X i)] (i : I) (snd : X i) : (ContinuousMap.sigmaMk i) snd = ⟨i, snd⟩

def ContinuousMap.sigmaMk {I : Type u_5} {X : I → Type u_7} [(i : I) → TopologicalSpace (X i)] (i : I) : C(X i, (i : I) × X i)

Sigma.mk i as a bundled continuous map.

Equations

• ContinuousMap.sigmaMk i = { toFun := Sigma.mk i, continuous_toFun := ⋯ }

@[simp]

theorem ContinuousMap.sigma_apply {I : Type u_5} {A : Type u_6} {X : I → Type u_7} [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] (f : (i : I) → C(X i, A)) (ig : (i : I) × X i) : (ContinuousMap.sigma f) ig = (f ig.fst) ig.snd

def ContinuousMap.sigma {I : Type u_5} {A : Type u_6} {X : I → Type u_7} [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] (f : (i : I) → C(X i, A)) : C((i : I) × X i, A)

To give a continuous map out of a disjoint union, it suffices to give a continuous map out of each term. This is Sigma.uncurry for continuous maps.

Equations

• ContinuousMap.sigma f = { toFun := fun (ig : (i : I) × X i) => (f ig.fst) ig.snd, continuous_toFun := ⋯ }

@[simp]

theorem ContinuousMap.sigmaEquiv_apply {I : Type u_5} (A : Type u_6) (X : I → Type u_7) [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] (f : (i : I) → C(X i, A)) : (ContinuousMap.sigmaEquiv A X) f = ContinuousMap.sigma f

@[simp]

theorem ContinuousMap.sigmaEquiv_symm_apply {I : Type u_5} (A : Type u_6) (X : I → Type u_7) [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] (f : C((i : I) × X i, A)) (i : I) : (ContinuousMap.sigmaEquiv A X).symm f i = f.comp (ContinuousMap.sigmaMk i)

def ContinuousMap.sigmaEquiv {I : Type u_5} (A : Type u_6) (X : I → Type u_7) [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] : ((i : I) → C(X i, A)) ≃ C((i : I) × X i, A)

Giving a continuous map out of a disjoint union is the same as giving a continuous map out of each term. This is a version of Equiv.piCurry for continuous maps.

Equations

• ContinuousMap.sigmaEquiv A X = { toFun := ContinuousMap.sigma, invFun := fun (f : C((i : I) × X i, A)) (i : I) => f.comp (ContinuousMap.sigmaMk i), left_inv := ⋯, right_inv := ⋯ }

def ContinuousMap.pi {I : Type u_5} {A : Type u_6} {X : I → Type u_7} [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] (f : (i : I) → C(A, X i)) : C(A, (i : I) → X i)

Abbreviation for product of continuous maps, which is continuous

Equations

• ContinuousMap.pi f = { toFun := fun (a : A) (i : I) => (f i) a, continuous_toFun := ⋯ }

@[simp]

theorem ContinuousMap.pi_eval {I : Type u_5} {A : Type u_6} {X : I → Type u_7} [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] (f : (i : I) → C(A, X i)) (a : A) : (ContinuousMap.pi f) a = fun (i : I) => (f i) a

@[simp]

theorem ContinuousMap.eval_apply {I : Type u_5} {X : I → Type u_7} [(i : I) → TopologicalSpace (X i)] (i : I) : ⇑(ContinuousMap.eval i) = Function.eval i

def ContinuousMap.eval {I : Type u_5} {X : I → Type u_7} [(i : I) → TopologicalSpace (X i)] (i : I) : C((j : I) → X j, X i)

Evaluation at point as a bundled continuous map.

Equations

• ContinuousMap.eval i = { toFun := Function.eval i, continuous_toFun := ⋯ }

@[simp]

theorem ContinuousMap.piEquiv_apply {I : Type u_5} (A : Type u_6) (X : I → Type u_7) [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] (f : (i : I) → C(A, X i)) : (ContinuousMap.piEquiv A X) f = ContinuousMap.pi f

@[simp]

theorem ContinuousMap.piEquiv_symm_apply {I : Type u_5} (A : Type u_6) (X : I → Type u_7) [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] (f : C(A, (i : I) → X i)) (i : I) : (ContinuousMap.piEquiv A X).symm f i = (ContinuousMap.eval i).comp f

def ContinuousMap.piEquiv {I : Type u_5} (A : Type u_6) (X : I → Type u_7) [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] : ((i : I) → C(A, X i)) ≃ C(A, (i : I) → X i)

Giving a continuous map out of a disjoint union is the same as giving a continuous map out of each term

Equations

• ContinuousMap.piEquiv A X = { toFun := ContinuousMap.pi, invFun := fun (f : C(A, (i : I) → X i)) (i : I) => (ContinuousMap.eval i).comp f, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem ContinuousMap.piMap_apply {I : Type u_5} {X : I → Type u_7} {Y : I → Type u_8} [(i : I) → TopologicalSpace (X i)] [(i : I) → TopologicalSpace (Y i)] (f : (i : I) → C(X i, Y i)) (a : (i : I) → X i) (i : I) : (ContinuousMap.piMap f) a i = (f i) (a i)

def ContinuousMap.piMap {I : Type u_5} {X : I → Type u_7} {Y : I → Type u_8} [(i : I) → TopologicalSpace (X i)] [(i : I) → TopologicalSpace (Y i)] (f : (i : I) → C(X i, Y i)) : C((i : I) → X i, (i : I) → Y i)

Combine a collection of bundled continuous maps C(X i, Y i) into a bundled continuous map C(∀ i, X i, ∀ i, Y i).

Equations

• ContinuousMap.piMap f = ContinuousMap.pi fun (i : I) => (f i).comp (ContinuousMap.eval i)

def ContinuousMap.precomp {I : Type u_5} {X : I → Type u_7} [(i : I) → TopologicalSpace (X i)] {ι : Type u_9} (φ : ι → I) : C((i : I) → X i, (i : ι) → X (φ i))

"Precomposition" as a continuous map between dependent types.

Equations

• ContinuousMap.precomp φ = { toFun := fun (f : (i : I) → X i) (j : ι) => f (φ j), continuous_toFun := ⋯ }

def ContinuousMap.restrict {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (s : Set α) (f : C(α, β)) : C(↑s, β)

The restriction of a continuous function α → β to a subset s of α.

Equations

• ContinuousMap.restrict s f = { toFun := ⇑f ∘ Subtype.val, continuous_toFun := ⋯ }

@[simp]

theorem ContinuousMap.coe_restrict {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (s : Set α) (f : C(α, β)) : ⇑(ContinuousMap.restrict s f) = ⇑f ∘ Subtype.val

@[simp]

theorem ContinuousMap.restrict_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (s : Set α) (x : ↑s) : (ContinuousMap.restrict s f) x = f ↑x

@[simp]

theorem ContinuousMap.restrict_apply_mk {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (s : Set α) (x : α) (hx : x ∈ s) : (ContinuousMap.restrict s f) ⟨x, hx⟩ = f x

theorem ContinuousMap.injective_restrict {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [T2Space β] {s : Set α} (hs : Dense s) : Function.Injective (ContinuousMap.restrict s)

@[simp]

theorem ContinuousMap.restrictPreimage_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (s : Set β) : ∀ (a : ↑(⇑f ⁻¹' s)), (f.restrictPreimage s) a = s.restrictPreimage (⇑f) a

def ContinuousMap.restrictPreimage {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (s : Set β) : C(↑(⇑f ⁻¹' s), ↑s)

The restriction of a continuous map to the preimage of a set.

Equations

• f.restrictPreimage s = { toFun := s.restrictPreimage ⇑f, continuous_toFun := ⋯ }

noncomputable def ContinuousMap.liftCover {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {ι : Type u_5} (S : ι → Set α) (φ : (i : ι) → C(↑(S i), β)) (hφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), (φ i) ⟨x, hxi⟩ = (φ j) ⟨x, hxj⟩) (hS : ∀ (x : α), ∃ (i : ι), S i ∈ nhds x) : C(α, β)

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).

Equations

• ContinuousMap.liftCover S φ hφ hS = { toFun := Set.liftCover S (fun (i : ι) => ⇑(φ i)) hφ ⋯, continuous_toFun := ⋯ }

@[simp]

theorem ContinuousMap.liftCover_coe {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {ι : Type u_5} {S : ι → Set α} {φ : (i : ι) → C(↑(S i), β)} {hφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), (φ i) ⟨x, hxi⟩ = (φ j) ⟨x, hxj⟩} {hS : ∀ (x : α), ∃ (i : ι), S i ∈ nhds x} {i : ι} (x : ↑(S i)) : (ContinuousMap.liftCover S φ hφ hS) ↑x = (φ i) x

theorem ContinuousMap.liftCover_restrict {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {ι : Type u_5} {S : ι → Set α} {φ : (i : ι) → C(↑(S i), β)} {hφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), (φ i) ⟨x, hxi⟩ = (φ j) ⟨x, hxj⟩} {hS : ∀ (x : α), ∃ (i : ι), S i ∈ nhds x} {i : ι} : ContinuousMap.restrict (S i) (ContinuousMap.liftCover S φ hφ hS) = φ i

noncomputable def ContinuousMap.liftCover' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (A : Set (Set α)) (F : (s : Set α) → s ∈ A → C(↑s, β)) (hF : ∀ (s : Set α) (hs : s ∈ A) (t : Set α) (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) : C(α, β)

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).

Equations

• ContinuousMap.liftCover' A F hF hA = let S := Subtype.val; let F_1 := fun (i : ↑A) => F ↑i ⋯; ContinuousMap.liftCover S F_1 ⋯ ⋯

@[simp]

theorem ContinuousMap.liftCover_coe' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {A : Set (Set α)} {F : (s : Set α) → s ∈ A → C(↑s, β)} {hF : ∀ (s : Set α) (hs : s ∈ A) (t : Set α) (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} {s : Set α} {hs : s ∈ A} (x : ↑s) : (ContinuousMap.liftCover' A F hF hA) ↑x = (F s hs) x

@[simp]

theorem ContinuousMap.liftCover_restrict' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {A : Set (Set α)} {F : (s : Set α) → s ∈ A → C(↑s, β)} {hF : ∀ (s : Set α) (hs : s ∈ A) (t : Set α) (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} {s : Set α} {hs : s ∈ A} : ContinuousMap.restrict s (ContinuousMap.liftCover' A F hF hA) = F s hs

@[simp]

theorem Function.RightInverse.homeomorph_symm_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : C(X, Y)} {f' : C(Y, X)} (hf : Function.RightInverse ⇑f' ⇑f) (b : Y) : hf.homeomorph.symm b = Quotient.mk'' (f' b)

@[simp]

theorem Function.RightInverse.homeomorph_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : C(X, Y)} {f' : C(Y, X)} (hf : Function.RightInverse ⇑f' ⇑f) (a : Quotient (Setoid.ker ⇑f)) : hf.homeomorph a = a.liftOn' ⇑f ⋯

def Function.RightInverse.homeomorph {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : C(X, Y)} {f' : C(Y, X)} (hf : Function.RightInverse ⇑f' ⇑f) : Quotient (Setoid.ker ⇑f) ≃ₜ Y

Setoid.quotientKerEquivOfRightInverse as a homeomorphism.

Equations

• hf.homeomorph = { toEquiv := Setoid.quotientKerEquivOfRightInverse (⇑f) (⇑f') hf, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem QuotientMap.homeomorph_symm_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : C(X, Y)} (hf : QuotientMap ⇑f) (b : Y) : hf.homeomorph.symm b = Quotient.mk'' (Function.surjInv ⋯ b)

@[simp]

theorem QuotientMap.homeomorph_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : C(X, Y)} (hf : QuotientMap ⇑f) (a : Quotient (Setoid.ker ⇑f)) : hf.homeomorph a = a.liftOn' ⇑f ⋯

noncomputable def QuotientMap.homeomorph {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : C(X, Y)} (hf : QuotientMap ⇑f) : Quotient (Setoid.ker ⇑f) ≃ₜ Y

The homeomorphism from the quotient of a quotient map to its codomain. This is Setoid.quotientKerEquivOfSurjective as a homeomorphism.

Equations

• hf.homeomorph = { toEquiv := Setoid.quotientKerEquivOfSurjective ⇑f ⋯, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem QuotientMap.lift_apply {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} (hf : QuotientMap ⇑f) (g : C(X, Z)) (h : Function.FactorsThrough ⇑g ⇑f) : ∀ (a : Y), (hf.lift g h) a = ((fun (i : Quotient (Setoid.ker ⇑f)) => i.liftOn' ⇑g ⋯) ∘ ⇑hf.homeomorph.symm) a

noncomputable def QuotientMap.lift {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} (hf : QuotientMap ⇑f) (g : C(X, Z)) (h : Function.FactorsThrough ⇑g ⇑f) : C(Y, Z)

Descend a continuous map, which is constant on the fibres, along a quotient map.

Equations

• hf.lift g h = { toFun := (fun (i : Quotient (Setoid.ker ⇑f)) => i.liftOn' ⇑g ⋯) ∘ ⇑hf.homeomorph.symm, continuous_toFun := ⋯ }

@[simp]

theorem QuotientMap.lift_comp {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} (hf : QuotientMap ⇑f) (g : C(X, Z)) (h : Function.FactorsThrough ⇑g ⇑f) : (hf.lift g h).comp f = g

The obvious triangle induced by QuotientMap.lift commutes:

     g
  X --→ Z
  |   ↗
f |  / hf.lift g h
  v /
  Y

@[simp]

theorem QuotientMap.liftEquiv_apply {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} (hf : QuotientMap ⇑f) (g : { g : C(X, Z) // Function.FactorsThrough ⇑g ⇑f }) : hf.liftEquiv g = hf.lift ↑g ⋯

@[simp]

theorem QuotientMap.liftEquiv_symm_apply_coe {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} (hf : QuotientMap ⇑f) (g : C(Y, Z)) : ↑(hf.liftEquiv.symm g) = g.comp f

noncomputable def QuotientMap.liftEquiv {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} (hf : QuotientMap ⇑f) : { g : C(X, Z) // Function.FactorsThrough ⇑g ⇑f } ≃ C(Y, Z)

QuotientMap.lift as an equivalence.

Equations

• hf.liftEquiv = { toFun := fun (g : { g : C(X, Z) // Function.FactorsThrough ⇑g ⇑f }) => hf.lift ↑g ⋯, invFun := fun (g : C(Y, Z)) => ⟨g.comp f, ⋯⟩, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem Homeomorph.toContinuousMap_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α ≃ₜ β) (a : α) : e.toContinuousMap a = e a

def Homeomorph.toContinuousMap {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α ≃ₜ β) : C(α, β)

The forward direction of a homeomorphism, as a bundled continuous map.

Equations

• e.toContinuousMap = { toFun := ⇑e, continuous_toFun := ⋯ }

instance Homeomorph.instCoeContinuousMap {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : Coe (α ≃ₜ β) C(α, β)

Homeomorph.toContinuousMap as a coercion.

Equations

• Homeomorph.instCoeContinuousMap = { coe := Homeomorph.toContinuousMap }

@[simp]

theorem Homeomorph.coe_refl {α : Type u_1} [TopologicalSpace α] : (Homeomorph.refl α).toContinuousMap = ContinuousMap.id α

@[simp]

theorem Homeomorph.coe_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : α ≃ₜ β) (g : β ≃ₜ γ) : (f.trans g).toContinuousMap = g.toContinuousMap.comp f.toContinuousMap

@[simp]

theorem Homeomorph.symm_comp_toContinuousMap {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) : f.symm.toContinuousMap.comp f.toContinuousMap = ContinuousMap.id α

Left inverse to a continuous map from a homeomorphism, mirroring Equiv.symm_comp_self.

@[simp]

theorem Homeomorph.toContinuousMap_comp_symm {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) : f.toContinuousMap.comp f.symm.toContinuousMap = ContinuousMap.id β

Right inverse to a continuous map from a homeomorphism, mirroring Equiv.self_comp_symm.