fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
noncomputable ofTopologicalSpace (X : Type*) [TopologicalSpace X] [CompactSpace X]
[T2Space X] : Compactum where
A := X
a := Ultrafilter.lim
unit := by
ext x
exact lim_eq (pure_le_nhds _)
assoc := by
ext FF
change Ultrafilter (Ultrafilter X) at FF
set x := (Ultrafilter.map Ultrafilter.lim FF).lim with c1
have c2 : ∀ (U : Set X) (F : Ultrafilter X), F.lim ∈ U → IsOpen U → U ∈ F := by
intro U F h1 hU
exact isOpen_iff_ultrafilter.mp hU _ h1 _ (Ultrafilter.le_nhds_lim _)
have c3 : ↑(Ultrafilter.map Ultrafilter.lim FF) ≤ 𝓝 x := by
rw [le_nhds_iff]
intro U hx hU
exact mem_coe.2 (c2 _ _ (by rwa [← c1]) hU)
have c4 : ∀ U : Set X, x ∈ U → IsOpen U → { G : Ultrafilter X | U ∈ G } ∈ FF := by
intro U hx hU
suffices Ultrafilter.lim ⁻¹' U ∈ FF by
apply mem_of_superset this
intro P hP
exact c2 U P hP hU
exact @c3 U (IsOpen.mem_nhds hU hx)
apply lim_eq
rw [le_nhds_iff]
exact c4 | def | Topology | [
"Mathlib.CategoryTheory.Monad.Types",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.CategoryTheory.Equivalence",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Data.Set.Constructions"
] | Mathlib/Topology/Category/Compactum.lean | ofTopologicalSpace | Given any compact Hausdorff space, we construct a Compactum. |
homOfContinuous {X Y : Compactum} (f : X → Y) (cont : Continuous f) : X ⟶ Y :=
{ f
h := by
rw [continuous_iff_ultrafilter] at cont
ext (F : Ultrafilter X)
specialize cont (X.str F) F (le_nhds_of_str_eq F (X.str F) rfl)
simp only [types_comp_apply]
exact str_eq_of_le_nhds (Ultrafilter.map f F) _ cont } | def | Topology | [
"Mathlib.CategoryTheory.Monad.Types",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.CategoryTheory.Equivalence",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Data.Set.Constructions"
] | Mathlib/Topology/Category/Compactum.lean | homOfContinuous | Any continuous map between Compacta is a morphism of compacta. |
compactumToCompHaus : Compactum ⥤ CompHaus where
obj X := { toTop := TopCat.of X, prop := trivial }
map := fun f => CompHausLike.ofHom _
{ toFun := f
continuous_toFun := Compactum.continuous_of_hom _ } | def | Topology | [
"Mathlib.CategoryTheory.Monad.Types",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.CategoryTheory.Equivalence",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Data.Set.Constructions"
] | Mathlib/Topology/Category/Compactum.lean | compactumToCompHaus | The functor functor from Compactum to CompHaus. |
full : compactumToCompHaus.{u}.Full where
map_surjective f := ⟨Compactum.homOfContinuous f.1 f.hom.2, rfl⟩ | instance | Topology | [
"Mathlib.CategoryTheory.Monad.Types",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.CategoryTheory.Equivalence",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Data.Set.Constructions"
] | Mathlib/Topology/Category/Compactum.lean | full | The functor `compactumToCompHaus` is full. |
faithful : compactumToCompHaus.Faithful where
map_injective := by
intro _ _ _ _ h
apply Monad.Algebra.Hom.ext
apply congrArg (fun f => f.hom.toFun) h | instance | Topology | [
"Mathlib.CategoryTheory.Monad.Types",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.CategoryTheory.Equivalence",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Data.Set.Constructions"
] | Mathlib/Topology/Category/Compactum.lean | faithful | The functor `compactumToCompHaus` is faithful. |
noncomputable isoOfTopologicalSpace {D : CompHaus} :
compactumToCompHaus.obj (Compactum.ofTopologicalSpace D) ≅ D where
hom := CompHausLike.ofHom _
{ toFun := id
continuous_toFun :=
continuous_def.2 fun _ h => by
rw [isOpen_iff_ultrafilter'] at h
exact h }
inv := CompHausLike.ofHom _
{ toFun := id
continuous_toFun :=
continuous_def.2 fun _ h1 => by
rw [isOpen_iff_ultrafilter']
intro _ h2
exact h1 _ h2 } | def | Topology | [
"Mathlib.CategoryTheory.Monad.Types",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.CategoryTheory.Equivalence",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Data.Set.Constructions"
] | Mathlib/Topology/Category/Compactum.lean | isoOfTopologicalSpace | This definition is used to prove essential surjectivity of `compactumToCompHaus`. |
essSurj : compactumToCompHaus.EssSurj :=
{ mem_essImage := fun X => ⟨Compactum.ofTopologicalSpace X, ⟨isoOfTopologicalSpace⟩⟩ } | instance | Topology | [
"Mathlib.CategoryTheory.Monad.Types",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.CategoryTheory.Equivalence",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Data.Set.Constructions"
] | Mathlib/Topology/Category/Compactum.lean | essSurj | The functor `compactumToCompHaus` is essentially surjective. |
isEquivalence : compactumToCompHaus.IsEquivalence where | instance | Topology | [
"Mathlib.CategoryTheory.Monad.Types",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.CategoryTheory.Equivalence",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Data.Set.Constructions"
] | Mathlib/Topology/Category/Compactum.lean | isEquivalence | The functor `compactumToCompHaus` is an equivalence of categories. |
compactumToCompHausCompForget :
compactumToCompHaus ⋙ CategoryTheory.forget CompHaus ≅ Compactum.forget :=
NatIso.ofComponents fun _ => eqToIso rfl
/-
TODO: `forget CompHaus` is monadic, as it is isomorphic to the composition
of an equivalence with the monadic functor `forget Compactum`.
Once we have the API to transfer monadicity of functors along such isomorphisms,
the instance `CreatesLimits (forget CompHaus)` can be deduced from this
monadicity.
-/ | def | Topology | [
"Mathlib.CategoryTheory.Monad.Types",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.CategoryTheory.Equivalence",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Data.Set.Constructions"
] | Mathlib/Topology/Category/Compactum.lean | compactumToCompHausCompForget | The forgetful functors of `Compactum` and `CompHaus` are compatible via
`compactumToCompHaus`. |
noncomputable CompHaus.forgetCreatesLimits : CreatesLimits (forget CompHaus) := by
let e : forget CompHaus ≅ compactumToCompHaus.inv ⋙ Compactum.forget :=
(((forget CompHaus).leftUnitor.symm ≪≫
Functor.isoWhiskerRight compactumToCompHaus.asEquivalence.symm.unitIso (forget CompHaus)) ≪≫
compactumToCompHaus.inv.associator compactumToCompHaus (forget CompHaus)) ≪≫
Functor.isoWhiskerLeft _ compactumToCompHausCompForget
exact createsLimitsOfNatIso e.symm | instance | Topology | [
"Mathlib.CategoryTheory.Monad.Types",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.CategoryTheory.Equivalence",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Data.Set.Constructions"
] | Mathlib/Topology/Category/Compactum.lean | CompHaus.forgetCreatesLimits | null |
noncomputable Profinite.forgetCreatesLimits : CreatesLimits (forget Profinite) := by
change CreatesLimits (profiniteToCompHaus ⋙ forget _)
infer_instance | instance | Topology | [
"Mathlib.CategoryTheory.Monad.Types",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.CategoryTheory.Equivalence",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Data.Set.Constructions"
] | Mathlib/Topology/Category/Compactum.lean | Profinite.forgetCreatesLimits | null |
DeltaGenerated where
/-- the underlying topological space -/
toTop : TopCat.{u}
/-- The underlying topological space is delta-generated. -/
deltaGenerated : DeltaGeneratedSpace toTop := by infer_instance | structure | Topology | [
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Compactness.DeltaGeneratedSpace"
] | Mathlib/Topology/Category/DeltaGenerated.lean | DeltaGenerated | The type of delta-generated topological spaces. |
of (X : Type u) [TopologicalSpace X] [DeltaGeneratedSpace X] : DeltaGenerated.{u} where
toTop := TopCat.of X
deltaGenerated := ‹_› | abbrev | Topology | [
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Compactness.DeltaGeneratedSpace"
] | Mathlib/Topology/Category/DeltaGenerated.lean | of | Constructor for objects of the category `DeltaGenerated` |
@[simps!]
deltaGeneratedToTop : DeltaGenerated.{u} ⥤ TopCat.{u} :=
inducedFunctor _ | def | Topology | [
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Compactness.DeltaGeneratedSpace"
] | Mathlib/Topology/Category/DeltaGenerated.lean | deltaGeneratedToTop | The forgetful functor `DeltaGenerated ⥤ TopCat` |
fullyFaithfulDeltaGeneratedToTop : deltaGeneratedToTop.{u}.FullyFaithful :=
fullyFaithfulInducedFunctor _ | def | Topology | [
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Compactness.DeltaGeneratedSpace"
] | Mathlib/Topology/Category/DeltaGenerated.lean | fullyFaithfulDeltaGeneratedToTop | `deltaGeneratedToTop` is fully faithful. |
@[simps!]
topToDeltaGenerated : TopCat.{u} ⥤ DeltaGenerated.{u} where
obj X := of (DeltaGeneratedSpace.of X)
map {_ Y} f := TopCat.ofHom ⟨f, (continuous_to_deltaGenerated (Y := Y)).mpr <|
continuous_le_dom deltaGenerated_le f.hom.continuous⟩ | def | Topology | [
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Compactness.DeltaGeneratedSpace"
] | Mathlib/Topology/Category/DeltaGenerated.lean | topToDeltaGenerated | The faithful (but not full) functor taking each topological space to its delta-generated
coreflection. |
coreflectorAdjunction : deltaGeneratedToTop ⊣ topToDeltaGenerated :=
Adjunction.mkOfUnitCounit {
unit := {
app X := TopCat.ofHom ⟨id, continuous_iff_coinduced_le.mpr (eq_deltaGenerated (X := X)).le⟩ }
counit := {
app X := TopCat.ofHom ⟨DeltaGeneratedSpace.counit, DeltaGeneratedSpace.continuous_counit⟩ }} | def | Topology | [
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Compactness.DeltaGeneratedSpace"
] | Mathlib/Topology/Category/DeltaGenerated.lean | coreflectorAdjunction | The adjunction between the forgetful functor `DeltaGenerated ⥤ TopCat` and its coreflector. |
deltaGeneratedToTop.coreflective : Coreflective deltaGeneratedToTop where
R := topToDeltaGenerated
adj := coreflectorAdjunction | instance | Topology | [
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Compactness.DeltaGeneratedSpace"
] | Mathlib/Topology/Category/DeltaGenerated.lean | deltaGeneratedToTop.coreflective | The category of delta-generated spaces is coreflective in the category of topological spaces. |
noncomputable deltaGeneratedToTop.createsColimits : CreatesColimits deltaGeneratedToTop :=
comonadicCreatesColimits deltaGeneratedToTop | instance | Topology | [
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Compactness.DeltaGeneratedSpace"
] | Mathlib/Topology/Category/DeltaGenerated.lean | deltaGeneratedToTop.createsColimits | null |
hasLimits : Limits.HasLimits DeltaGenerated :=
hasLimits_of_coreflective deltaGeneratedToTop | instance | Topology | [
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Compactness.DeltaGeneratedSpace"
] | Mathlib/Topology/Category/DeltaGenerated.lean | hasLimits | null |
hasColimits : Limits.HasColimits DeltaGenerated :=
hasColimits_of_hasColimits_createsColimits deltaGeneratedToTop | instance | Topology | [
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Compactness.DeltaGeneratedSpace"
] | Mathlib/Topology/Category/DeltaGenerated.lean | hasColimits | null |
FinTopCat where
/-- carrier of a finite topological space. -/
toTop : TopCat.{u} -- TODO: turn this into an `extends`?
[fintype : Fintype toTop] | structure | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.TopCat.Basic"
] | Mathlib/Topology/Category/FinTopCat.lean | FinTopCat | A bundled finite topological space. |
of (X : Type u) [Fintype X] [TopologicalSpace X] : FinTopCat where
toTop := TopCat.of X
fintype := ‹_›
@[simp] | def | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.TopCat.Basic"
] | Mathlib/Topology/Category/FinTopCat.lean | of | Construct a bundled `FinTopCat` from the underlying type and the appropriate typeclasses. |
coe_of (X : Type u) [Fintype X] [TopologicalSpace X] :
(of X : Type u) = X :=
rfl | theorem | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.TopCat.Basic"
] | Mathlib/Topology/Category/FinTopCat.lean | coe_of | null |
Locale :=
Frmᵒᵖ deriving LargeCategory | def | Topology | [
"Mathlib.Order.Category.Frm",
"Mathlib.Topology.Category.CompHaus.Frm"
] | Mathlib/Topology/Category/Locale.lean | Locale | The category of locales. |
of (α : Type*) [Frame α] : Locale :=
op <| Frm.of α
@[simp] | def | Topology | [
"Mathlib.Order.Category.Frm",
"Mathlib.Topology.Category.CompHaus.Frm"
] | Mathlib/Topology/Category/Locale.lean | of | Construct a bundled `Locale` from a `Frame`. |
coe_of (α : Type*) [Frame α] : ↥(of α) = α :=
rfl | theorem | Topology | [
"Mathlib.Order.Category.Frm",
"Mathlib.Topology.Category.CompHaus.Frm"
] | Mathlib/Topology/Category/Locale.lean | coe_of | null |
@[simps!]
topToLocale : TopCat ⥤ Locale :=
topCatOpToFrm.rightOp | def | Topology | [
"Mathlib.Order.Category.Frm",
"Mathlib.Topology.Category.CompHaus.Frm"
] | Mathlib/Topology/Category/Locale.lean | topToLocale | The forgetful functor from `Top` to `Locale` which forgets that the space has "enough points". |
CompHausToLocale.faithful : (compHausToTop ⋙ topToLocale.{u}).Faithful :=
⟨fun h => by
dsimp at h
exact ConcreteCategory.ext (Opens.comap_injective (congr_arg Frm.Hom.hom
(Quiver.Hom.op_inj h)))⟩ | instance | Topology | [
"Mathlib.Order.Category.Frm",
"Mathlib.Topology.Category.CompHaus.Frm"
] | Mathlib/Topology/Category/Locale.lean | CompHausToLocale.faithful | null |
Sequential where
/-- The underlying topological space of an object of `Sequential`. -/
toTop : TopCat.{u} -- TODO: turn this into `extends`
/-- The underlying topological space is sequential. -/
[is_sequential : SequentialSpace toTop] | structure | Topology | [
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.Sequences",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.Category.TopCat.Basic"
] | Mathlib/Topology/Category/Sequential.lean | Sequential | The type sequential topological spaces. |
of : Sequential.{u} where
toTop := TopCat.of X
is_sequential := ‹_› | abbrev | Topology | [
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.Sequences",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.Category.TopCat.Basic"
] | Mathlib/Topology/Category/Sequential.lean | of | Constructor for objects of the category `Sequential`. |
@[simps!]
sequentialToTop : Sequential.{u} ⥤ TopCat.{u} :=
inducedFunctor _ | def | Topology | [
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.Sequences",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.Category.TopCat.Basic"
] | Mathlib/Topology/Category/Sequential.lean | sequentialToTop | The fully faithful embedding of `Sequential` in `TopCat`. |
fullyFaithfulSequentialToTop : sequentialToTop.FullyFaithful :=
fullyFaithfulInducedFunctor _ | def | Topology | [
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.Sequences",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.Category.TopCat.Basic"
] | Mathlib/Topology/Category/Sequential.lean | fullyFaithfulSequentialToTop | The functor to `TopCat` is indeed fully faithful. |
@[simps hom inv]
isoOfHomeo {X Y : Sequential.{u}} (f : X ≃ₜ Y) : X ≅ Y where
hom := TopCat.ofHom ⟨f, f.continuous⟩
inv := TopCat.ofHom ⟨f.symm, f.symm.continuous⟩
hom_inv_id := by
ext x
exact f.symm_apply_apply x
inv_hom_id := by
ext x
exact f.apply_symm_apply x | def | Topology | [
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.Sequences",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.Category.TopCat.Basic"
] | Mathlib/Topology/Category/Sequential.lean | isoOfHomeo | Construct an isomorphism from a homeomorphism. |
@[simps]
homeoOfIso {X Y : Sequential.{u}} (f : X ≅ Y) : X ≃ₜ Y where
toFun := f.hom
invFun := f.inv
left_inv := f.hom_inv_id_apply
right_inv := f.inv_hom_id_apply
continuous_toFun := f.hom.hom.continuous
continuous_invFun := f.inv.hom.continuous | def | Topology | [
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.Sequences",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.Category.TopCat.Basic"
] | Mathlib/Topology/Category/Sequential.lean | homeoOfIso | Construct a homeomorphism from an isomorphism. |
@[simps]
isoEquivHomeo {X Y : Sequential.{u}} : (X ≅ Y) ≃ (X ≃ₜ Y) where
toFun := homeoOfIso
invFun := isoOfHomeo | def | Topology | [
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.Sequences",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.Category.TopCat.Basic"
] | Mathlib/Topology/Category/Sequential.lean | isoEquivHomeo | The equivalence between isomorphisms in `Sequential` and homeomorphisms
of topological spaces. |
TopCommRingCat where
/-- carrier of a topological commutative ring. -/
α : Type u
[isCommRing : CommRing α]
[isTopologicalSpace : TopologicalSpace α]
[isTopologicalRing : IsTopologicalRing α] | structure | Topology | [
"Mathlib.Algebra.Category.Ring.Basic",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Category/TopCommRingCat.lean | TopCommRingCat | A bundled topological commutative ring. |
of (X : Type u) [CommRing X] [TopologicalSpace X] [IsTopologicalRing X] : TopCommRingCat :=
⟨X⟩ | abbrev | Topology | [
"Mathlib.Algebra.Category.Ring.Basic",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Category/TopCommRingCat.lean | of | Construct a bundled `TopCommRingCat` from the underlying type and the appropriate typeclasses. |
coe_of (X : Type u) [CommRing X] [TopologicalSpace X] [IsTopologicalRing X] :
(of X : Type u) = X := rfl | theorem | Topology | [
"Mathlib.Algebra.Category.Ring.Basic",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Category/TopCommRingCat.lean | coe_of | null |
hasForgetToCommRingCat : HasForget₂ TopCommRingCat CommRingCat :=
HasForget₂.mk' (fun R => CommRingCat.of R) (fun _ => rfl)
(fun f => CommRingCat.ofHom f.val) HEq.rfl | instance | Topology | [
"Mathlib.Algebra.Category.Ring.Basic",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Category/TopCommRingCat.lean | hasForgetToCommRingCat | null |
forgetToCommRingCatTopologicalSpace (R : TopCommRingCat) :
TopologicalSpace ((forget₂ TopCommRingCat CommRingCat).obj R) :=
R.isTopologicalSpace | instance | Topology | [
"Mathlib.Algebra.Category.Ring.Basic",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Category/TopCommRingCat.lean | forgetToCommRingCatTopologicalSpace | null |
hasForgetToTopCat : HasForget₂ TopCommRingCat TopCat :=
HasForget₂.mk' (fun R => TopCat.of R) (fun _ => rfl) (fun f => TopCat.ofHom ⟨⇑f.1, f.2⟩) HEq.rfl | instance | Topology | [
"Mathlib.Algebra.Category.Ring.Basic",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Category/TopCommRingCat.lean | hasForgetToTopCat | The forgetful functor to `TopCat`. |
forgetToTopCatCommRing (R : TopCommRingCat) :
CommRing ((forget₂ TopCommRingCat TopCat).obj R) :=
R.isCommRing | instance | Topology | [
"Mathlib.Algebra.Category.Ring.Basic",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Category/TopCommRingCat.lean | forgetToTopCatCommRing | null |
forgetToTopCatTopologicalRing (R : TopCommRingCat) :
IsTopologicalRing ((forget₂ TopCommRingCat TopCat).obj R) :=
R.isTopologicalRing | instance | Topology | [
"Mathlib.Algebra.Category.Ring.Basic",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Category/TopCommRingCat.lean | forgetToTopCatTopologicalRing | null |
UniformSpaceCat : Type (u + 1) where
/-- The underlying uniform space. -/
carrier : Type u
[str : UniformSpace carrier]
attribute [instance] UniformSpaceCat.str | structure | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | UniformSpaceCat | An object in the category of uniform spaces. |
of (α : Type u) [UniformSpace α] : UniformSpaceCat where
carrier := α | abbrev | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | of | Construct a bundled `UniformSpace` from the underlying type and the typeclass. |
@[ext]
Hom (X Y : UniformSpaceCat) where
/-- The underlying `UniformContinuous` function. -/
hom' : { f : X → Y // UniformContinuous f } | structure | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | Hom | A bundled uniform continuous map. |
instFunLike (X Y : UniformSpaceCat) :
FunLike { f : X → Y // UniformContinuous f } X Y where
coe := Subtype.val
coe_injective' _ _ h := Subtype.ext h | instance | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | instFunLike | null |
Hom.hom {X Y : UniformSpaceCat} (f : Hom X Y) :=
ConcreteCategory.hom (C := UniformSpaceCat) f | abbrev | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | Hom.hom | Turn a morphism in `UniformSpaceCat` back into a function which is `UniformContinuous`. |
ofHom {X Y : Type u} [UniformSpace X] [UniformSpace Y]
(f : { f : X → Y // UniformContinuous f }) : of X ⟶ of Y :=
ConcreteCategory.ofHom f | abbrev | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | ofHom | Typecheck a function which is `UniformContinuous` as a morphism in `UniformSpaceCat`. |
coe_of (X : Type u) [UniformSpace X] : (of X : Type u) = X :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | coe_of | null |
hom_comp {X Y Z : UniformSpaceCat} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g).hom = ⟨g ∘ f, g.hom.prop.comp f.hom.prop⟩ :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | hom_comp | null |
hom_id (X : UniformSpaceCat) : (𝟙 X : X ⟶ X).hom = ⟨id, uniformContinuous_id⟩ :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | hom_id | null |
hom_ofHom {X Y : Type u} [UniformSpace X] [UniformSpace Y]
(f : { f : X → Y // UniformContinuous f }) : (ofHom f).hom = f :=
rfl | theorem | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | hom_ofHom | null |
coe_comp {X Y Z : UniformSpaceCat} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g : X → Z) = g ∘ f :=
rfl | theorem | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | coe_comp | null |
coe_id (X : UniformSpaceCat) : (𝟙 X : X → X) = id :=
rfl | theorem | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | coe_id | null |
coe_mk {X Y : UniformSpaceCat} (f : X → Y) (hf : UniformContinuous f) :
(⟨f, hf⟩ : X ⟶ Y).hom = f :=
rfl
@[ext] | theorem | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | coe_mk | null |
hom_ext {X Y : UniformSpaceCat} {f g : X ⟶ Y} (h : (f : X → Y) = g) : f = g :=
Hom.ext (Subtype.ext h) | theorem | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | hom_ext | null |
hasForgetToTop : HasForget₂ UniformSpaceCat.{u} TopCat.{u} where
forget₂ :=
{ obj := fun X => TopCat.of X
map := fun f => TopCat.ofHom
{ toFun := f
continuous_toFun := f.hom.property.continuous } } | instance | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | hasForgetToTop | The forgetful functor from uniform spaces to topological spaces. |
CpltSepUniformSpace where
/-- The underlying space -/
α : Type u
[isUniformSpace : UniformSpace α]
[isCompleteSpace : CompleteSpace α]
[isT0 : T0Space α] | structure | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | CpltSepUniformSpace | A (bundled) complete separated uniform space. |
toUniformSpace (X : CpltSepUniformSpace) : UniformSpaceCat :=
UniformSpaceCat.of X | def | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | toUniformSpace | The function forgetting that a complete separated uniform spaces is complete and separated. |
completeSpace (X : CpltSepUniformSpace) : CompleteSpace (toUniformSpace X).carrier :=
CpltSepUniformSpace.isCompleteSpace X | instance | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | completeSpace | null |
t0Space (X : CpltSepUniformSpace) : T0Space (toUniformSpace X).carrier :=
CpltSepUniformSpace.isT0 X | instance | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | t0Space | null |
of (X : Type u) [UniformSpace X] [CompleteSpace X] [T0Space X] : CpltSepUniformSpace :=
⟨X⟩
@[simp] | def | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | of | Construct a bundled `UniformSpace` from the underlying type and the appropriate typeclasses. |
coe_of (X : Type u) [UniformSpace X] [CompleteSpace X] [T0Space X] :
(of X : Type u) = X :=
rfl | theorem | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | coe_of | null |
category : LargeCategory CpltSepUniformSpace :=
InducedCategory.category toUniformSpace | instance | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | category | The category instance on `CpltSepUniformSpace`. |
instFunLike (X Y : CpltSepUniformSpace) :
FunLike { f : X → Y // UniformContinuous f } X Y where
coe := Subtype.val
coe_injective' _ _ h := Subtype.ext h | instance | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | instFunLike | null |
concreteCategory : ConcreteCategory CpltSepUniformSpace
({ f : · → · // UniformContinuous f }) :=
InducedCategory.concreteCategory toUniformSpace | instance | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | concreteCategory | The concrete category instance on `CpltSepUniformSpace`. |
hasForgetToUniformSpace : HasForget₂ CpltSepUniformSpace UniformSpaceCat :=
InducedCategory.hasForget₂ toUniformSpace
@[simp] | instance | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | hasForgetToUniformSpace | null |
hom_comp {X Y Z : CpltSepUniformSpace} (f : X ⟶ Y) (g : Y ⟶ Z) :
ConcreteCategory.hom (f ≫ g) = ⟨g ∘ f, g.hom.prop.comp f.hom.prop⟩ :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | hom_comp | null |
hom_id (X : CpltSepUniformSpace) :
ConcreteCategory.hom (𝟙 X : X ⟶ X) = ⟨id, uniformContinuous_id⟩ :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | hom_id | null |
hom_ofHom {X Y : Type u} [UniformSpace X] [UniformSpace Y]
(f : { f : X → Y // UniformContinuous f }) : (UniformSpaceCat.ofHom f).hom = f :=
rfl | theorem | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | hom_ofHom | null |
@[simps map]
noncomputable completionFunctor : UniformSpaceCat ⥤ CpltSepUniformSpace where
obj X := CpltSepUniformSpace.of (Completion X)
map f := ofHom ⟨Completion.map f.1, Completion.uniformContinuous_map⟩
map_id _ := hom_ext Completion.map_id
map_comp f g := hom_ext (Completion.map_comp g.hom.property f.hom.property).symm | def | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | completionFunctor | The functor turning uniform spaces into complete separated uniform spaces. |
completionHom (X : UniformSpaceCat) :
X ⟶ (forget₂ CpltSepUniformSpace UniformSpaceCat).obj (completionFunctor.obj X) where
hom'.val := ((↑) : X → Completion X)
hom'.property := Completion.uniformContinuous_coe X
@[simp] | def | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | completionHom | The inclusion of a uniform space into its completion. |
completionHom_val (X : UniformSpaceCat) (x) : (completionHom X) x = (x : Completion X) :=
rfl | theorem | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | completionHom_val | null |
noncomputable extensionHom {X : UniformSpaceCat} {Y : CpltSepUniformSpace}
(f : X ⟶ (forget₂ CpltSepUniformSpace UniformSpaceCat).obj Y) :
completionFunctor.obj X ⟶ Y where
hom'.val := Completion.extension f
hom'.property := Completion.uniformContinuous_extension
@[simp] | def | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | extensionHom | The mate of a morphism from a `UniformSpace` to a `CpltSepUniformSpace`. |
extensionHom_val {X : UniformSpaceCat} {Y : CpltSepUniformSpace}
(f : X ⟶ (forget₂ _ _).obj Y) (x) : (extensionHom f) x = Completion.extension f x :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | extensionHom_val | null |
extension_comp_coe {X : UniformSpaceCat} {Y : CpltSepUniformSpace}
(f : toUniformSpace (CpltSepUniformSpace.of (Completion X)) ⟶ toUniformSpace Y) :
extensionHom (completionHom X ≫ f) = f := by
ext x
exact congr_fun (Completion.extension_comp_coe f.hom.property) x | theorem | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | extension_comp_coe | null |
noncomputable adj : completionFunctor ⊣ forget₂ CpltSepUniformSpace UniformSpaceCat :=
Adjunction.mkOfHomEquiv
{ homEquiv := fun X Y =>
{ toFun := fun f => completionHom X ≫ f
invFun := fun f => extensionHom f
left_inv := fun f => by dsimp; rw [extension_comp_coe]
right_inv := fun f => by
ext x
rcases f with ⟨⟨_, _⟩⟩
exact @Completion.extension_coe _ _ _ _ _ (CpltSepUniformSpace.t0Space _)
‹_› _ }
homEquiv_naturality_left_symm := fun {X' X Y} f g => by
ext x
dsimp [-Function.comp_apply]
erw [Completion.extension_map (γ := Y) g.hom.2 f.hom.2]
rfl } | def | Topology | [
"Mathlib.CategoryTheory.Adjunction.Reflective",
"Mathlib.CategoryTheory.Monad.Limits",
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/Category/UniformSpace.lean | adj | The completion functor is left adjoint to the forgetful functor. |
ultrafilterBasis (α : Type u) : Set (Set (Ultrafilter α)) :=
range fun s : Set α ↦ { u | s ∈ u }
variable {α : Type u} | def | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | ultrafilterBasis | Basis for the topology on `Ultrafilter α`. |
Ultrafilter.topologicalSpace : TopologicalSpace (Ultrafilter α) :=
TopologicalSpace.generateFrom (ultrafilterBasis α) | instance | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | Ultrafilter.topologicalSpace | null |
ultrafilterBasis_is_basis : TopologicalSpace.IsTopologicalBasis (ultrafilterBasis α) :=
⟨by
rintro _ ⟨a, rfl⟩ _ ⟨b, rfl⟩ u ⟨ua, ub⟩
refine ⟨_, ⟨a ∩ b, rfl⟩, inter_mem ua ub, fun v hv ↦ ⟨?_, ?_⟩⟩ <;> apply mem_of_superset hv <;>
simp [inter_subset_right],
eq_univ_of_univ_subset <| subset_sUnion_of_mem <| ⟨univ, eq_univ_of_forall fun _ ↦ univ_mem⟩,
rfl⟩ | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | ultrafilterBasis_is_basis | null |
ultrafilter_isOpen_basic (s : Set α) : IsOpen { u : Ultrafilter α | s ∈ u } :=
ultrafilterBasis_is_basis.isOpen ⟨s, rfl⟩ | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | ultrafilter_isOpen_basic | The basic open sets for the topology on ultrafilters are open. |
ultrafilter_isClosed_basic (s : Set α) : IsClosed { u : Ultrafilter α | s ∈ u } := by
rw [← isOpen_compl_iff]
convert ultrafilter_isOpen_basic sᶜ using 1
ext u
exact Ultrafilter.compl_mem_iff_notMem.symm | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | ultrafilter_isClosed_basic | The basic open sets for the topology on ultrafilters are also closed. |
ultrafilter_converges_iff {u : Ultrafilter (Ultrafilter α)} {x : Ultrafilter α} :
↑u ≤ 𝓝 x ↔ x = joinM u := by
rw [eq_comm, ← Ultrafilter.coe_le_coe]
change ↑u ≤ 𝓝 x ↔ ∀ s ∈ x, { v : Ultrafilter α | s ∈ v } ∈ u
simp only [TopologicalSpace.nhds_generateFrom, le_iInf_iff, ultrafilterBasis, le_principal_iff,
mem_setOf_eq]
constructor
· intro h a ha
exact h _ ⟨ha, a, rfl⟩
· rintro h a ⟨xi, a, rfl⟩
exact h _ xi | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | ultrafilter_converges_iff | Every ultrafilter `u` on `Ultrafilter α` converges to a unique
point of `Ultrafilter α`, namely `joinM u`. |
ultrafilter_compact : CompactSpace (Ultrafilter α) :=
⟨isCompact_iff_ultrafilter_le_nhds.mpr fun f _ ↦
⟨joinM f, trivial, ultrafilter_converges_iff.mpr rfl⟩⟩ | instance | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | ultrafilter_compact | null |
Ultrafilter.t2Space : T2Space (Ultrafilter α) :=
t2_iff_ultrafilter.mpr fun {x y} f fx fy ↦
have hx : x = joinM f := ultrafilter_converges_iff.mp fx
have hy : y = joinM f := ultrafilter_converges_iff.mp fy
hx.trans hy.symm | instance | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | Ultrafilter.t2Space | null |
@[simp] Ultrafilter.tendsto_pure_self (b : Ultrafilter α) : Tendsto pure b (𝓝 b) := by
rw [Tendsto, ← coe_map, ultrafilter_converges_iff]
ext s
change s ∈ b ↔ {t | s ∈ t} ∈ map pure b
simp_rw [mem_map, preimage_setOf_eq, mem_pure, setOf_mem_eq] | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | Ultrafilter.tendsto_pure_self | null |
ultrafilter_comap_pure_nhds (b : Ultrafilter α) : comap pure (𝓝 b) ≤ b := by
rw [TopologicalSpace.nhds_generateFrom]
simp only [comap_iInf, comap_principal]
intro s hs
rw [← le_principal_iff]
refine iInf_le_of_le { u | s ∈ u } ?_
refine iInf_le_of_le ⟨hs, ⟨s, rfl⟩⟩ ?_
exact principal_mono.2 fun _ ↦ id | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | ultrafilter_comap_pure_nhds | null |
denseRange_pure : DenseRange (pure : α → Ultrafilter α) :=
fun x ↦ mem_closure_iff_ultrafilter.mpr
⟨x.map pure, range_mem_map, ultrafilter_converges_iff.mpr (bind_pure x).symm⟩ | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | denseRange_pure | The range of `pure : α → Ultrafilter α` is dense in `Ultrafilter α`. |
induced_topology_pure :
TopologicalSpace.induced (pure : α → Ultrafilter α) Ultrafilter.topologicalSpace = ⊥ := by
apply eq_bot_of_singletons_open
intro x
use { u : Ultrafilter α | {x} ∈ u }, ultrafilter_isOpen_basic _
simp | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | induced_topology_pure | The map `pure : α → Ultrafilter α` induces on `α` the discrete topology. |
isDenseInducing_pure : @IsDenseInducing _ _ ⊥ _ (pure : α → Ultrafilter α) :=
letI : TopologicalSpace α := ⊥
⟨⟨induced_topology_pure.symm⟩, denseRange_pure⟩ | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | isDenseInducing_pure | `pure : α → Ultrafilter α` defines a dense inducing of `α` in `Ultrafilter α`. |
isDenseEmbedding_pure : @IsDenseEmbedding _ _ ⊥ _ (pure : α → Ultrafilter α) :=
letI : TopologicalSpace α := ⊥
{ isDenseInducing_pure with injective := Ultrafilter.pure_injective } | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | isDenseEmbedding_pure | `pure : α → Ultrafilter α` defines a dense embedding of `α` in `Ultrafilter α`. |
Ultrafilter.extend (f : α → γ) : Ultrafilter α → γ :=
letI : TopologicalSpace α := ⊥
isDenseInducing_pure.extend f
variable [T2Space γ]
@[simp] | def | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | Ultrafilter.extend | The extension of a function `α → γ` to a function `Ultrafilter α → γ`.
When `γ` is a compact Hausdorff space it will be continuous. |
ultrafilter_extend_extends (f : α → γ) : Ultrafilter.extend f ∘ pure = f := by
letI : TopologicalSpace α := ⊥
haveI : DiscreteTopology α := ⟨rfl⟩
exact funext (isDenseInducing_pure.extend_eq continuous_of_discreteTopology)
@[simp] | lemma | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | ultrafilter_extend_extends | null |
ultrafilter_extend_pure (f : α → γ) (a : α) : Ultrafilter.extend f (pure a) = f a :=
congr_fun (ultrafilter_extend_extends f) a
variable [CompactSpace γ] | lemma | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | ultrafilter_extend_pure | null |
continuous_ultrafilter_extend (f : α → γ) : Continuous (Ultrafilter.extend f) := by
have h (b : Ultrafilter α) : ∃ c, Tendsto f (comap pure (𝓝 b)) (𝓝 c) :=
let ⟨c, _, h'⟩ :=
isCompact_univ.ultrafilter_le_nhds (b.map f) (by rw [le_principal_iff]; exact univ_mem)
⟨c, le_trans (map_mono (ultrafilter_comap_pure_nhds _)) h'⟩
let _ : TopologicalSpace α := ⊥
exact isDenseInducing_pure.continuous_extend h | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | continuous_ultrafilter_extend | null |
ultrafilter_extend_eq_iff {f : α → γ} {b : Ultrafilter α} {c : γ} :
Ultrafilter.extend f b = c ↔ ↑(b.map f) ≤ 𝓝 c :=
⟨fun h ↦ by
let b' : Ultrafilter (Ultrafilter α) := b.map pure
have t : ↑b' ≤ 𝓝 b := ultrafilter_converges_iff.mpr (bind_pure _).symm
rw [← h]
have := (continuous_ultrafilter_extend f).tendsto b
refine le_trans ?_ (le_trans (map_mono t) this)
change _ ≤ map (Ultrafilter.extend f ∘ pure) ↑b
rw [ultrafilter_extend_extends]
exact le_rfl,
fun h ↦
let _ : TopologicalSpace α := ⊥
isDenseInducing_pure.extend_eq_of_tendsto
(le_trans (map_mono (ultrafilter_comap_pure_nhds _)) h)⟩ | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | ultrafilter_extend_eq_iff | The value of `Ultrafilter.extend f` on an ultrafilter `b` is the
unique limit of the ultrafilter `b.map f` in `γ`. |
PreStoneCech : Type u :=
Quot fun F G : Ultrafilter α ↦ ∃ x, (F : Filter α) ≤ 𝓝 x ∧ (G : Filter α) ≤ 𝓝 x
variable {α} | def | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | PreStoneCech | Auxiliary construction towards the Stone-Čech compactification of a topological space.
It should not be used after the Stone-Čech compactification is constructed. |
preStoneCechUnit (x : α) : PreStoneCech α :=
Quot.mk _ (pure x : Ultrafilter α) | def | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | preStoneCechUnit | The natural map from α to its pre-Stone-Čech compactification. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.