File: Mathlib/Topology/Category/Sequential.lean

full_name: Sequential.is_sequential
code: /-- The type sequential topological spaces. -/
structure Sequential where
  /-- The underlying topological space of an object of `Sequential`. -/
  toTop : TopCat.{u}
  /-- The underlying topological space is sequential. -/
  [is_sequential : SequentialSpace toTop]
goal: SequentialSpace ↑self.toTop

full_name: Sequential.toTop
code: /-- The type sequential topological spaces. -/
structure Sequential where
  /-- The underlying topological space of an object of `Sequential`. -/
  toTop : TopCat.{u}
  /-- The underlying topological space is sequential. -/
  [is_sequential : SequentialSpace toTop]
goal: TopCat

full_name: Sequential
code: /-- The type sequential topological spaces. -/
structure Sequential where
  /-- The underlying topological space of an object of `Sequential`. -/
  toTop : TopCat.{u}
  /-- The underlying topological space is sequential. -/
  [is_sequential : SequentialSpace toTop]
goal: Type (u + 1)

full_name: Sequential.instInhabited
code: instance : Inhabited Sequential.{u} 
goal: Inhabited Sequential

full_name: Sequential.instCoeSortType
code: instance : CoeSort Sequential Type* 
goal: CoeSort Sequential (Type u_1)

full_name: Sequential.instCategory
code: instance : Category.{u, u+1} Sequential.{u} 
goal: CategoryTheory.Category.{u, u + 1} Sequential

full_name: Sequential.instConcreteCategory
code: instance : ConcreteCategory.{u} Sequential.{u} 
goal: CategoryTheory.ConcreteCategory Sequential

full_name: Sequential.of
code: /-- Constructor for objects of the category `Sequential`. -/
def of : Sequential.{u} 
goal: Sequential

full_name: Sequential.sequentialToTop
code: /-- The fully faithful embedding of `Sequential` in `TopCat`. -/
@[simps!]
def sequentialToTop : Sequential.{u} ⥤ TopCat.{u} 
goal: Sequential ⥤ TopCat

full_name: Sequential.sequentialToTop_obj
code: /-- The fully faithful embedding of `Sequential` in `TopCat`. -/
@[simps!]
def sequentialToTop : Sequential.{u} ⥤ TopCat.{u} 
goal: Sequential.sequentialToTop.obj self = self.toTop

full_name: Sequential.sequentialToTop_map
code: /-- The fully faithful embedding of `Sequential` in `TopCat`. -/
@[simps!]
def sequentialToTop : Sequential.{u} ⥤ TopCat.{u} 
goal: ∀ {X Y : CategoryTheory.InducedCategory TopCat Sequential.toTop} (f : X ⟶ Y), Sequential.sequentialToTop.map f = f

full_name: Sequential.fullyFaithfulSequentialToTop
code: /-- The functor to `TopCat` is indeed fully faithful.-/
def fullyFaithfulSequentialToTop : sequentialToTop.FullyFaithful 
goal: Sequential.sequentialToTop.FullyFaithful

full_name: Sequential.instFullTopCatSequentialToTop
code: instance : sequentialToTop.{u}.Full  
goal: Sequential.sequentialToTop.Full

full_name: Sequential.instFaithfulTopCatSequentialToTop
code: instance : sequentialToTop.{u}.Faithful 
goal: Sequential.sequentialToTop.Faithful

full_name: Sequential.isoOfHomeo
code: /-- Construct an isomorphism from a homeomorphism. -/
@[simps hom inv]
def isoOfHomeo {X Y : Sequential.{u}} (f : X ≃ₜ Y) : X ≅ Y 
goal: X ≅ Y

full_name: Sequential.isoOfHomeo_inv
code: /-- Construct an isomorphism from a homeomorphism. -/
@[simps hom inv]
def isoOfHomeo {X Y : Sequential.{u}} (f : X ≃ₜ Y) : X ≅ Y 
goal: (Sequential.isoOfHomeo f).inv = { toFun := ⇑f.symm, continuous_toFun := ⋯ }

full_name: Sequential.isoOfHomeo_hom
code: /-- Construct an isomorphism from a homeomorphism. -/
@[simps hom inv]
def isoOfHomeo {X Y : Sequential.{u}} (f : X ≃ₜ Y) : X ≅ Y 
goal: (Sequential.isoOfHomeo f).hom = { toFun := ⇑f, continuous_toFun := ⋯ }

full_name: Sequential.homeoOfIso
code: /-- Construct a homeomorphism from an isomorphism. -/
@[simps]
def homeoOfIso {X Y : Sequential.{u}} (f : X ≅ Y) : X ≃ₜ Y 
goal: ↑X.toTop ≃ₜ ↑Y.toTop

full_name: Sequential.homeoOfIso_symm_apply
code: /-- Construct a homeomorphism from an isomorphism. -/
@[simps]
def homeoOfIso {X Y : Sequential.{u}} (f : X ≅ Y) : X ≃ₜ Y 
goal: (Sequential.homeoOfIso f).symm a = f.inv a

full_name: Sequential.homeoOfIso_apply
code: /-- Construct a homeomorphism from an isomorphism. -/
@[simps]
def homeoOfIso {X Y : Sequential.{u}} (f : X ≅ Y) : X ≃ₜ Y 
goal: (Sequential.homeoOfIso f) a = f.hom a

full_name: Sequential.isoEquivHomeo_apply
code: /-- The equivalence between isomorphisms in `Sequential` and homeomorphisms
of topological spaces. -/
@[simps]
def isoEquivHomeo {X Y : Sequential.{u}} : (X ≅ Y) ≃ (X ≃ₜ Y) 
goal: Sequential.isoEquivHomeo f = Sequential.homeoOfIso f

full_name: Sequential.isoEquivHomeo_symm_apply
code: /-- The equivalence between isomorphisms in `Sequential` and homeomorphisms
of topological spaces. -/
@[simps]
def isoEquivHomeo {X Y : Sequential.{u}} : (X ≅ Y) ≃ (X ≃ₜ Y) 
goal: Sequential.isoEquivHomeo.symm f = Sequential.isoOfHomeo f

full_name: Sequential.isoEquivHomeo
code: /-- The equivalence between isomorphisms in `Sequential` and homeomorphisms
of topological spaces. -/
@[simps]
def isoEquivHomeo {X Y : Sequential.{u}} : (X ≅ Y) ≃ (X ≃ₜ Y) 
goal: (X ≅ Y) ≃ (↑X.toTop ≃ₜ ↑Y.toTop)