Covering Maps

This file defines covering maps.

Main definitions

• IsEvenlyCovered f x I: A point x is evenly covered by f : E → X with fiber I if I is discrete and there is a Trivialization of f at x with fiber I.
• IsCoveringMap f: A function f : E → X is a covering map if every point x is evenly covered by f with fiber f ⁻¹' {x}. The fibers f ⁻¹' {x} must be discrete, but if X is not connected, then the fibers f ⁻¹' {x} are not necessarily isomorphic. Also, f is not assumed to be surjective, so the fibers are even allowed to be empty.

def IsEvenlyCovered {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] (f : E → X) (x : X) (I : Type u_3) [TopologicalSpace I] : Prop

A point x : X is evenly covered by f : E → X if x has an evenly covered neighborhood.

Equations

• IsEvenlyCovered f x I = (DiscreteTopology I ∧ ∃ (t : Trivialization I f), x ∈ t.baseSet)

noncomputable def IsEvenlyCovered.toTrivialization {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} {x : X} {I : Type u_3} [TopologicalSpace I] (h : IsEvenlyCovered f x I) : Trivialization (↑(f ⁻¹' {x})) f

If x is evenly covered by f, then we can construct a trivialization of f at x.

Equations

• h.toTrivialization = (Classical.choose ⋯).transFiberHomeomorph ((Classical.choose ⋯).preimageSingletonHomeomorph ⋯).symm

theorem IsEvenlyCovered.mem_toTrivialization_baseSet {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} {x : X} {I : Type u_3} [TopologicalSpace I] (h : IsEvenlyCovered f x I) : x ∈ h.toTrivialization.baseSet

theorem IsEvenlyCovered.toTrivialization_apply {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} {x : E} {I : Type u_3} [TopologicalSpace I] (h : IsEvenlyCovered f (f x) I) : (↑h.toTrivialization x).2 = ⟨x, ⋯⟩

theorem IsEvenlyCovered.continuousAt {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} {x : E} {I : Type u_3} [TopologicalSpace I] (h : IsEvenlyCovered f (f x) I) : ContinuousAt f x

theorem IsEvenlyCovered.to_isEvenlyCovered_preimage {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} {x : X} {I : Type u_3} [TopologicalSpace I] (h : IsEvenlyCovered f x I) : IsEvenlyCovered f x ↑(f ⁻¹' {x})

def IsCoveringMapOn {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] (f : E → X) (s : Set X) : Prop

A covering map is a continuous function f : E → X with discrete fibers such that each point of X has an evenly covered neighborhood.

Equations

• IsCoveringMapOn f s = ∀ x ∈ s, IsEvenlyCovered f x ↑(f ⁻¹' {x})

theorem IsCoveringMapOn.mk {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] (f : E → X) (s : Set X) (F : X → Type u_3) [(x : X) → TopologicalSpace (F x)] [hF : ∀ (x : X), DiscreteTopology (F x)] (e : (x : X) → x ∈ s → Trivialization (F x) f) (h : ∀ (x : X) (hx : x ∈ s), x ∈ (e x hx).baseSet) : IsCoveringMapOn f s

theorem IsCoveringMapOn.continuousAt {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} {s : Set X} (hf : IsCoveringMapOn f s) {x : E} (hx : f x ∈ s) : ContinuousAt f x

theorem IsCoveringMapOn.continuousOn {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} {s : Set X} (hf : IsCoveringMapOn f s) : ContinuousOn f (f ⁻¹' s)

theorem IsCoveringMapOn.isLocalHomeomorphOn {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} {s : Set X} (hf : IsCoveringMapOn f s) : IsLocalHomeomorphOn f (f ⁻¹' s)

def IsCoveringMap {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] (f : E → X) : Prop

A covering map is a continuous function f : E → X with discrete fibers such that each point of X has an evenly covered neighborhood.

Equations

• IsCoveringMap f = ∀ (x : X), IsEvenlyCovered f x ↑(f ⁻¹' {x})

theorem isCoveringMap_iff_isCoveringMapOn_univ {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} : IsCoveringMap f ↔ IsCoveringMapOn f Set.univ

theorem IsCoveringMap.isCoveringMapOn {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} (hf : IsCoveringMap f) : IsCoveringMapOn f Set.univ

theorem IsCoveringMap.mk {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] (f : E → X) (F : X → Type u_3) [(x : X) → TopologicalSpace (F x)] [∀ (x : X), DiscreteTopology (F x)] (e : (x : X) → Trivialization (F x) f) (h : ∀ (x : X), x ∈ (e x).baseSet) : IsCoveringMap f

theorem IsCoveringMap.continuous {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} (hf : IsCoveringMap f) : Continuous f

theorem IsCoveringMap.isLocalHomeomorph {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} (hf : IsCoveringMap f) : IsLocalHomeomorph f

theorem IsCoveringMap.isOpenMap {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} (hf : IsCoveringMap f) : IsOpenMap f

theorem IsCoveringMap.quotientMap {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} (hf : IsCoveringMap f) (hf' : Function.Surjective f) : QuotientMap f

theorem IsCoveringMap.isSeparatedMap {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} (hf : IsCoveringMap f) : IsSeparatedMap f

theorem IsCoveringMap.eq_of_comp_eq {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} (hf : IsCoveringMap f) {A : Type u_3} [TopologicalSpace A] {g₁ : A → E} {g₂ : A → E} [PreconnectedSpace A] (h₁ : Continuous g₁) (h₂ : Continuous g₂) (he : f ∘ g₁ = f ∘ g₂) (a : A) (ha : g₁ a = g₂ a) : g₁ = g₂

theorem IsCoveringMap.eqOn_of_comp_eqOn {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} (hf : IsCoveringMap f) {A : Type u_3} [TopologicalSpace A] {s : Set A} (hs : IsPreconnected s) {g₁ : A → E} {g₂ : A → E} (h₁ : ContinuousOn g₁ s) (h₂ : ContinuousOn g₂ s) (he : Set.EqOn (f ∘ g₁) (f ∘ g₂) s) {a : A} (has : a ∈ s) (ha : g₁ a = g₂ a) : Set.EqOn g₁ g₂ s

theorem IsCoveringMap.const_of_comp {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} (hf : IsCoveringMap f) {A : Type u_3} [TopologicalSpace A] {g : A → E} [PreconnectedSpace A] (cont : Continuous g) (he : ∀ (a a' : A), f (g a) = f (g a')) (a : A) (a' : A) : g a = g a'

theorem IsCoveringMap.constOn_of_comp {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} (hf : IsCoveringMap f) {A : Type u_3} [TopologicalSpace A] {s : Set A} (hs : IsPreconnected s) {g : A → E} (cont : ContinuousOn g s) (he : ∀ a ∈ s, ∀ a' ∈ s, f (g a) = f (g a')) {a : A} {a' : A} (ha : a ∈ s) (ha' : a' ∈ s) : g a = g a'

theorem IsFiberBundle.isCoveringMap {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} {F : Type u_3} [TopologicalSpace F] [DiscreteTopology F] (hf : ∀ (x : X), ∃ (e : Trivialization F f), x ∈ e.baseSet) : IsCoveringMap f

theorem FiberBundle.isCoveringMap {X : Type u_2} [TopologicalSpace X] {F : Type u_3} {E : X → Type u_4} [TopologicalSpace F] [DiscreteTopology F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : X) → TopologicalSpace (E x)] [FiberBundle F E] : IsCoveringMap Bundle.TotalSpace.proj