File: Mathlib/Topology/Category/CompactlyGenerated.lean

full_name: TopologicalSpace.compactlyGenerated
code: /--
The compactly generated topology on a topological space `X`. This is the finest topology
which makes all maps from compact Hausdorff spaces to `X`, which are continuous for the original
topology, continuous.

Note: this definition should be used with an explicit universe parameter `u` for the size of the
compact Hausdorff spaces mapping to `X`.
-/
def TopologicalSpace.compactlyGenerated (X : Type w) [TopologicalSpace X] : TopologicalSpace X 
goal: TopologicalSpace X

full_name: continuous_from_compactlyGenerated
code: lemma continuous_from_compactlyGenerated {X : Type w} [TopologicalSpace X]
    {Y : Type*} [t : TopologicalSpace Y] (f : X → Y)
      (h : ∀ (S : CompHaus.{u}) (g : C(S, X)), Continuous (f ∘ g)) :
        Continuous[compactlyGenerated.{u} X, t] f 
goal: Continuous f

full_name: UCompactlyGeneratedSpace
code: /--
A topological space `X` is compactly generated if its topology is finer than (and thus equal to)
the compactly generated topology, i.e. it is coinduced by the continuous maps from compact
Hausdorff spaces to `X`.
-/
class UCompactlyGeneratedSpace (X : Type w) [t : TopologicalSpace X] : Prop where
  /-- The topology of `X` is finer than the compactly generated topology. -/
  le_compactlyGenerated : t ≤ compactlyGenerated.{u} X
goal: Prop

full_name: UCompactlyGeneratedSpace.le_compactlyGenerated
code: /--
A topological space `X` is compactly generated if its topology is finer than (and thus equal to)
the compactly generated topology, i.e. it is coinduced by the continuous maps from compact
Hausdorff spaces to `X`.
-/
class UCompactlyGeneratedSpace (X : Type w) [t : TopologicalSpace X] : Prop where
  /-- The topology of `X` is finer than the compactly generated topology. -/
  le_compactlyGenerated : t ≤ compactlyGenerated.{u} X
goal: t ≤ TopologicalSpace.compactlyGenerated X

full_name: eq_compactlyGenerated
code: lemma eq_compactlyGenerated {X : Type w} [t : TopologicalSpace X] [UCompactlyGeneratedSpace.{u} X] :
    t = compactlyGenerated.{u} X 
goal: t = TopologicalSpace.compactlyGenerated X

full_name: instUCompactlyGeneratedSpaceOfDiscreteTopology
code: instance (X : Type w) [t : TopologicalSpace X] [DiscreteTopology X] :
    UCompactlyGeneratedSpace.{u} X 
goal: UCompactlyGeneratedSpace X

full_name: continuous_from_uCompactlyGeneratedSpace
code: lemma continuous_from_uCompactlyGeneratedSpace {X : Type w} [TopologicalSpace X]
    [UCompactlyGeneratedSpace.{u} X] {Y : Type*} [TopologicalSpace Y] (f : X → Y)
      (h : ∀ (S : CompHaus.{u}) (g : C(S, X)), Continuous (f ∘ g)) : Continuous f 
goal: Continuous f

full_name: uCompactlyGeneratedSpace_of_continuous_maps
code: lemma uCompactlyGeneratedSpace_of_continuous_maps {X : Type w} [t : TopologicalSpace X]
    (h : ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),
      (∀ (S : CompHaus.{u}) (g : C(S, X)), Continuous (f ∘ g)) → Continuous f) :
        UCompactlyGeneratedSpace.{u} X 
goal: UCompactlyGeneratedSpace X

full_name: CompactlyGenerated.toTop
code: /-- The type of `u`-compactly generated `w`-small topological spaces. -/
structure CompactlyGenerated where
  /-- The underlying topological space of an object of `CompactlyGenerated`. -/
  toTop : TopCat.{w}
  /-- The underlying topological space is compactly generated. -/
  [is_compactly_generated : UCompactlyGeneratedSpace.{u} toTop]
goal: TopCat

full_name: CompactlyGenerated
code: /-- The type of `u`-compactly generated `w`-small topological spaces. -/
structure CompactlyGenerated where
  /-- The underlying topological space of an object of `CompactlyGenerated`. -/
  toTop : TopCat.{w}
  /-- The underlying topological space is compactly generated. -/
  [is_compactly_generated : UCompactlyGeneratedSpace.{u} toTop]
goal: Type (w + 1)

full_name: CompactlyGenerated.is_compactly_generated
code: /-- The type of `u`-compactly generated `w`-small topological spaces. -/
structure CompactlyGenerated where
  /-- The underlying topological space of an object of `CompactlyGenerated`. -/
  toTop : TopCat.{w}
  /-- The underlying topological space is compactly generated. -/
  [is_compactly_generated : UCompactlyGeneratedSpace.{u} toTop]
goal: UCompactlyGeneratedSpace ↑self.toTop

full_name: CompactlyGenerated.instInhabited
code: instance : Inhabited CompactlyGenerated.{u, w} 
goal: Inhabited CompactlyGenerated

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

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

full_name: CompactlyGenerated.instConcreteCategory
code: instance : ConcreteCategory.{w} CompactlyGenerated.{u, w} 
goal: CategoryTheory.ConcreteCategory CompactlyGenerated

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

full_name: CompactlyGenerated.compactlyGeneratedToTop_obj
code: /-- The fully faithful embedding of `CompactlyGenerated` in `TopCat`. -/
@[simps!]
def compactlyGeneratedToTop : CompactlyGenerated.{u, w} ⥤ TopCat.{w} 
goal: CompactlyGenerated.compactlyGeneratedToTop.obj self = self.toTop

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

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

full_name: CompactlyGenerated.fullyFaithfulCompactlyGeneratedToTop
code: /-- `compactlyGeneratedToTop` is fully faithful. -/
def fullyFaithfulCompactlyGeneratedToTop : compactlyGeneratedToTop.{u, w}.FullyFaithful 
goal: CompactlyGenerated.compactlyGeneratedToTop.FullyFaithful

full_name: CompactlyGenerated.instFullTopCatCompactlyGeneratedToTop
code: instance : compactlyGeneratedToTop.{u, w}.Full 
goal: CompactlyGenerated.compactlyGeneratedToTop.Full

full_name: CompactlyGenerated.instFaithfulTopCatCompactlyGeneratedToTop
code: instance : compactlyGeneratedToTop.{u, w}.Faithful 
goal: CompactlyGenerated.compactlyGeneratedToTop.Faithful

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

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

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

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

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

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

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

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

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