File: Mathlib/Topology/Homotopy/Contractible.lean

full_name: ContinuousMap.Nullhomotopic
code: /-- A map is nullhomotopic if it is homotopic to a constant map. -/
def Nullhomotopic (f : C(X, Y)) : Prop 
goal: Prop

full_name: ContinuousMap.nullhomotopic_of_constant
code: theorem nullhomotopic_of_constant (y : Y) : Nullhomotopic (ContinuousMap.const X y) 
goal: (ContinuousMap.const X y).Nullhomotopic

full_name: ContinuousMap.Nullhomotopic.comp_right
code: theorem Nullhomotopic.comp_right {f : C(X, Y)} (hf : f.Nullhomotopic) (g : C(Y, Z)) :
    (g.comp f).Nullhomotopic 
goal: (g.comp f).Nullhomotopic

full_name: ContinuousMap.Nullhomotopic.comp_left
code: theorem Nullhomotopic.comp_left {f : C(Y, Z)} (hf : f.Nullhomotopic) (g : C(X, Y)) :
    (f.comp g).Nullhomotopic 
goal: (f.comp g).Nullhomotopic

full_name: ContractibleSpace.hequiv_unit'
code: /-- A contractible space is one that is homotopy equivalent to `Unit`. -/
class ContractibleSpace (X : Type*) [TopologicalSpace X] : Prop where
  hequiv_unit' : Nonempty (X ≃ₕ Unit)
goal: Nonempty (X ≃ₕ Unit)

full_name: ContractibleSpace
code: /-- A contractible space is one that is homotopy equivalent to `Unit`. -/
class ContractibleSpace (X : Type*) [TopologicalSpace X] : Prop where
  hequiv_unit' : Nonempty (X ≃ₕ Unit)
goal: Prop

full_name: ContractibleSpace.hequiv_unit
code: theorem ContractibleSpace.hequiv_unit (X : Type*) [TopologicalSpace X] [ContractibleSpace X] :
    Nonempty (X ≃ₕ Unit) 
goal: Nonempty (X ≃ₕ Unit)

full_name: id_nullhomotopic
code: theorem id_nullhomotopic (X : Type*) [TopologicalSpace X] [ContractibleSpace X] :
    (ContinuousMap.id X).Nullhomotopic 
goal: (ContinuousMap.id X).Nullhomotopic

full_name: contractible_iff_id_nullhomotopic
code: theorem contractible_iff_id_nullhomotopic (Y : Type*) [TopologicalSpace Y] :
    ContractibleSpace Y ↔ (ContinuousMap.id Y).Nullhomotopic 
goal: ContractibleSpace Y ↔ (ContinuousMap.id Y).Nullhomotopic

full_name: ContinuousMap.HomotopyEquiv.contractibleSpace
code: protected theorem ContinuousMap.HomotopyEquiv.contractibleSpace [ContractibleSpace Y] (e : X ≃ₕ Y) :
    ContractibleSpace X 
goal: ContractibleSpace X

full_name: ContinuousMap.HomotopyEquiv.contractibleSpace_iff
code: protected theorem ContinuousMap.HomotopyEquiv.contractibleSpace_iff (e : X ≃ₕ Y) :
    ContractibleSpace X ↔ ContractibleSpace Y 
goal: ContractibleSpace X ↔ ContractibleSpace Y

full_name: Homeomorph.contractibleSpace
code: protected theorem Homeomorph.contractibleSpace [ContractibleSpace Y] (e : X ≃ₜ Y) :
    ContractibleSpace X 
goal: ContractibleSpace X

full_name: Homeomorph.contractibleSpace_iff
code: protected theorem Homeomorph.contractibleSpace_iff (e : X ≃ₜ Y) :
    ContractibleSpace X ↔ ContractibleSpace Y 
goal: ContractibleSpace X ↔ ContractibleSpace Y

full_name: ContractibleSpace.instOfUnique
code: instance [Unique Y] : ContractibleSpace Y 
goal: ContractibleSpace Y

full_name: ContractibleSpace.hequiv
code: theorem hequiv [ContractibleSpace X] [ContractibleSpace Y] :
    Nonempty (X ≃ₕ Y) 
goal: Nonempty (X ≃ₕ Y)

full_name: ContractibleSpace.instPathConnectedSpace
code: instance (priority := 100) [ContractibleSpace X] : PathConnectedSpace X 
goal: PathConnectedSpace X