File: Mathlib/Topology/Covering.lean

full_name: IsEvenlyCovered
code: /-- A point `x : X` is evenly covered by `f : E → X` if `x` has an evenly covered neighborhood. -/
def IsEvenlyCovered (x : X) (I : Type*) [TopologicalSpace I] 
goal: Prop

full_name: IsEvenlyCovered.toTrivialization
code: /-- If `x` is evenly covered by `f`, then we can construct a trivialization of `f` at `x`. -/
noncomputable def toTrivialization {x : X} {I : Type*} [TopologicalSpace I]
    (h : IsEvenlyCovered f x I) : Trivialization (f ⁻¹' {x}) f 
goal: Trivialization (↑(f ⁻¹' {x})) f

full_name: IsEvenlyCovered.mem_toTrivialization_baseSet
code: theorem mem_toTrivialization_baseSet {x : X} {I : Type*} [TopologicalSpace I]
    (h : IsEvenlyCovered f x I) : x ∈ h.toTrivialization.baseSet 
goal: x ∈ h.toTrivialization.baseSet

full_name: IsEvenlyCovered.toTrivialization_apply
code: theorem toTrivialization_apply {x : E} {I : Type*} [TopologicalSpace I]
    (h : IsEvenlyCovered f (f x) I) : (h.toTrivialization x).2 = ⟨x, rfl⟩ 
goal: (↑h.toTrivialization x).2 = ⟨x, ⋯⟩

full_name: IsEvenlyCovered.continuousAt
code: protected theorem continuousAt {x : E} {I : Type*} [TopologicalSpace I]
    (h : IsEvenlyCovered f (f x) I) : ContinuousAt f x 
goal: ContinuousAt f x

full_name: IsEvenlyCovered.to_isEvenlyCovered_preimage
code: theorem to_isEvenlyCovered_preimage {x : X} {I : Type*} [TopologicalSpace I]
    (h : IsEvenlyCovered f x I) : IsEvenlyCovered f x (f ⁻¹' {x}) 
goal: IsEvenlyCovered f x ↑(f ⁻¹' {x})

full_name: IsCoveringMapOn
code: /-- A covering map is a continuous function `f : E → X` with discrete fibers such that each point
  of `X` has an evenly covered neighborhood. -/
def IsCoveringMapOn 
goal: Prop

full_name: IsCoveringMapOn.mk
code: theorem mk (F : X → Type*) [∀ x, TopologicalSpace (F x)] [hF : ∀ x, DiscreteTopology (F x)]
    (e : ∀ x ∈ s, Trivialization (F x) f) (h : ∀ (x : X) (hx : x ∈ s), x ∈ (e x hx).baseSet) :
    IsCoveringMapOn f s 
goal: IsCoveringMapOn f s

full_name: IsCoveringMapOn.continuousAt
code: protected theorem continuousAt (hf : IsCoveringMapOn f s) {x : E} (hx : f x ∈ s) :
    ContinuousAt f x 
goal: ContinuousAt f x

full_name: IsCoveringMapOn.continuousOn
code: protected theorem continuousOn (hf : IsCoveringMapOn f s) : ContinuousOn f (f ⁻¹' s) 
goal: ContinuousOn f (f ⁻¹' s)

full_name: IsCoveringMapOn.isLocalHomeomorphOn
code: protected theorem isLocalHomeomorphOn (hf : IsCoveringMapOn f s) :
    IsLocalHomeomorphOn f (f ⁻¹' s) 
goal: IsLocalHomeomorphOn f (f ⁻¹' s)

full_name: IsCoveringMap
code: /-- A covering map is a continuous function `f : E → X` with discrete fibers such that each point
  of `X` has an evenly covered neighborhood. -/
def IsCoveringMap 
goal: Prop

full_name: isCoveringMap_iff_isCoveringMapOn_univ
code: theorem isCoveringMap_iff_isCoveringMapOn_univ : IsCoveringMap f ↔ IsCoveringMapOn f Set.univ 
goal: IsCoveringMap f ↔ IsCoveringMapOn f Set.univ

full_name: IsCoveringMap.isCoveringMapOn
code: protected theorem IsCoveringMap.isCoveringMapOn (hf : IsCoveringMap f) :
    IsCoveringMapOn f Set.univ 
goal: IsCoveringMapOn f Set.univ

full_name: IsCoveringMap.mk
code: theorem mk (F : X → Type*) [∀ x, TopologicalSpace (F x)] [∀ x, DiscreteTopology (F x)]
    (e : ∀ x, Trivialization (F x) f) (h : ∀ x, x ∈ (e x).baseSet) : IsCoveringMap f 
goal: IsCoveringMap f

full_name: IsCoveringMap.continuous
code: protected theorem continuous : Continuous f 
goal: Continuous f

full_name: IsCoveringMap.isLocalHomeomorph
code: protected theorem isLocalHomeomorph : IsLocalHomeomorph f 
goal: IsLocalHomeomorph f

full_name: IsCoveringMap.isOpenMap
code: protected theorem isOpenMap : IsOpenMap f 
goal: IsOpenMap f

full_name: IsCoveringMap.quotientMap
code: protected theorem quotientMap (hf' : Function.Surjective f) : QuotientMap f 
goal: QuotientMap f

full_name: IsCoveringMap.isSeparatedMap
code: protected theorem isSeparatedMap : IsSeparatedMap f 
goal: IsSeparatedMap f

full_name: IsCoveringMap.eq_of_comp_eq
code: theorem eq_of_comp_eq [PreconnectedSpace A] (h₁ : Continuous g₁) (h₂ : Continuous g₂)
    (he : f ∘ g₁ = f ∘ g₂) (a : A) (ha : g₁ a = g₂ a) : g₁ = g₂ 
goal: g₁ = g₂

full_name: IsCoveringMap.eqOn_of_comp_eqOn
code: theorem eqOn_of_comp_eqOn (h₁ : ContinuousOn g₁ s) (h₂ : ContinuousOn g₂ s)
    (he : s.EqOn (f ∘ g₁) (f ∘ g₂)) {a : A} (has : a ∈ s) (ha : g₁ a = g₂ a) : s.EqOn g₁ g₂ 
goal: Set.EqOn g₁ g₂ s

full_name: IsCoveringMap.const_of_comp
code: theorem const_of_comp [PreconnectedSpace A] (cont : Continuous g)
    (he : ∀ a a', f (g a) = f (g a')) (a a') : g a = g a' 
goal: g a = g a'

full_name: IsCoveringMap.constOn_of_comp
code: theorem constOn_of_comp (cont : ContinuousOn g s)
    (he : ∀ a ∈ s, ∀ a' ∈ s, f (g a) = f (g a'))
    {a a'} (ha : a ∈ s) (ha' : a' ∈ s) : g a = g a' 
goal: g a = g a'

full_name: IsFiberBundle.isCoveringMap
code: protected theorem IsFiberBundle.isCoveringMap {F : Type*} [TopologicalSpace F] [DiscreteTopology F]
    (hf : ∀ x : X, ∃ e : Trivialization F f, x ∈ e.baseSet) : IsCoveringMap f 
goal: IsCoveringMap f

full_name: FiberBundle.isCoveringMap
code: protected theorem FiberBundle.isCoveringMap {F : Type*} {E : X → Type*} [TopologicalSpace F]
    [DiscreteTopology F] [TopologicalSpace (Bundle.TotalSpace F E)] [∀ x, TopologicalSpace (E x)]
    [FiberBundle F E] : IsCoveringMap (π F E) 
goal: IsCoveringMap Bundle.TotalSpace.proj