statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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constructibleTopology (X : Type*) [TopologicalSpace X] : TopologicalSpace X | .generateFrom (constructibleTopologySubbasis X) | def | constructibleTopology | Topology.Spectral | Mathlib/Topology/Spectral/ConstructibleTopology.lean | [] | [
"TopologicalSpace",
"constructibleTopologySubbasis"
] | The constructible topology on a topological space `X` has as a subbasis
the open and compact sets of `X` and their complements. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
WithConstructibleTopology (X : Type*) [TopologicalSpace X] : Type _ | WithTopology X (constructibleTopology X) | abbrev | WithConstructibleTopology | Topology.Spectral | Mathlib/Topology/Spectral/ConstructibleTopology.lean | [] | [
"TopologicalSpace",
"WithTopology",
"constructibleTopology"
] | A type synonym for `X` that is equipped with the constructible topology of `X`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
WithConstructibleTopology.isOpen_iff {s : Set (WithConstructibleTopology X)} :
IsOpen s ↔ IsOpen[constructibleTopology X] (WithTopology.toTopology _ ⁻¹' s) | WithTopology.isOpen_iff _ | lemma | WithConstructibleTopology.isOpen_iff | Topology.Spectral | Mathlib/Topology/Spectral/ConstructibleTopology.lean | [] | [
"IsOpen",
"Set",
"WithConstructibleTopology",
"WithTopology.isOpen_iff",
"constructibleTopology"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
WithConstructibleTopology.isClosed_iff {s : Set (WithConstructibleTopology X)} :
IsClosed s ↔ IsClosed[constructibleTopology X] (WithTopology.toTopology _ ⁻¹' s) | WithTopology.isClosed_iff _ | lemma | WithConstructibleTopology.isClosed_iff | Topology.Spectral | Mathlib/Topology/Spectral/ConstructibleTopology.lean | [] | [
"IsClosed",
"Set",
"WithConstructibleTopology",
"WithTopology.isClosed_iff",
"constructibleTopology"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.isOpen_constructibleTopology_of_isOpen {s : Set X}
(hs : IsCompact s) (ho : IsOpen s) : IsOpen[constructibleTopology X] s | by
apply TopologicalSpace.isOpen_generateFrom_of_mem
simp [constructibleTopologySubbasis, ho, hs] | lemma | IsCompact.isOpen_constructibleTopology_of_isOpen | Topology.Spectral | Mathlib/Topology/Spectral/ConstructibleTopology.lean | [] | [
"IsCompact",
"IsOpen",
"Set",
"TopologicalSpace.isOpen_generateFrom_of_mem",
"constructibleTopology",
"constructibleTopologySubbasis"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.isOpen_constructibleTopology_of_isClosed {s : Set X}
(hs : IsCompact sᶜ) (ho : IsClosed s) : IsOpen[constructibleTopology X] s | by
apply TopologicalSpace.isOpen_generateFrom_of_mem
simp [constructibleTopologySubbasis, ho, hs] | lemma | IsCompact.isOpen_constructibleTopology_of_isClosed | Topology.Spectral | Mathlib/Topology/Spectral/ConstructibleTopology.lean | [] | [
"IsClosed",
"IsCompact",
"IsOpen",
"Set",
"TopologicalSpace.isOpen_generateFrom_of_mem",
"constructibleTopology",
"constructibleTopologySubbasis"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compl_mem_constructibleTopologySubbasis_iff {s : Set X} :
sᶜ ∈ constructibleTopologySubbasis X ↔ s ∈ constructibleTopologySubbasis X | by
grind [constructibleTopologySubbasis, isClosed_compl_iff, compl_compl] | lemma | compl_mem_constructibleTopologySubbasis_iff | Topology.Spectral | Mathlib/Topology/Spectral/ConstructibleTopology.lean | [] | [
"Set",
"compl_compl",
"constructibleTopologySubbasis",
"isClosed_compl_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isCompact_of_mem_constructibleTopologySubbasis [CompactSpace X] {s : Set X}
(hs : s ∈ constructibleTopologySubbasis X) : IsCompact s | by
obtain (hs | hs) := hs
· exact hs.2
· exact hs.1.isCompact | lemma | isCompact_of_mem_constructibleTopologySubbasis | Topology.Spectral | Mathlib/Topology/Spectral/ConstructibleTopology.lean | [] | [
"CompactSpace",
"IsCompact",
"Set",
"constructibleTopologySubbasis"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isCompact_sInter_of_subset_constructibleTopologySubbasis [CompactSpace X]
[QuasiSeparatedSpace X] {s : Set (Set X)} (hs : s ⊆ constructibleTopologySubbasis X)
(hf : s.Finite) : IsCompact (⋂₀ s) | by
refine QuasiSeparatedSpace.isCompact_sInter hf (fun t ht ↦ ?_) (fun t ht ↦ ?_)
· apply (hs ht).imp <;> grind
· exact isCompact_of_mem_constructibleTopologySubbasis (hs ht) | lemma | isCompact_sInter_of_subset_constructibleTopologySubbasis | Topology.Spectral | Mathlib/Topology/Spectral/ConstructibleTopology.lean | [] | [
"CompactSpace",
"IsCompact",
"QuasiSeparatedSpace",
"QuasiSeparatedSpace.isCompact_sInter",
"Set",
"constructibleTopologySubbasis",
"isCompact_of_mem_constructibleTopologySubbasis"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compactSpace_withConstructibleTopology [CompactSpace X] [QuasiSober X]
[PrespectralSpace X] [QuasiSeparatedSpace X] :
CompactSpace (WithConstructibleTopology X) | by
suffices h : @CompactSpace X (constructibleTopology X) from
@Function.Surjective.compactSpace _ _ (constructibleTopology X) _ _
(WithTopology.continuous_toTopology _) _ (WithTopology.toTopology_surjective _)
let 𝔅 := constructibleTopologySubbasis X
/- It suffices to check that any subset of `𝔅` for... | instance | compactSpace_withConstructibleTopology | Topology.Spectral | Mathlib/Topology/Spectral/ConstructibleTopology.lean | [] | [
"CompactSpace",
"CompactSpace.nonempty_sInter",
"Function.Surjective.compactSpace",
"IsClosed",
"IsCompact",
"IsCompact.nonempty_inter_sInter",
"IsIrreducible",
"PrespectralSpace",
"PrespectralSpace.exists_isClosed_of_not_isPreirreducible",
"QuasiSeparatedSpace",
"QuasiSober",
"Set",
"Set.di... | If `X` is quasi-separated, quasi-sober, prespectral and quasi-compact, then `X`
is still quasi-compact in the constructible topology. This holds in particular for spectral
spaces. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsSpectralMap (f : α → β) : Prop extends Continuous f where
/-- A function between topological spaces is spectral if it is continuous and the preimage of
every compact open set is compact open. -/
isCompact_preimage_of_isOpen ⦃s : Set β⦄ : IsOpen s → IsCompact s → IsCompact (f ⁻¹' s) | structure | IsSpectralMap | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"Continuous",
"IsCompact",
"IsOpen",
"Set"
] | A function between topological spaces is spectral if it is continuous and the preimage of every
compact open set is compact open. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.preimage_of_isOpen (hf : IsSpectralMap f) (h₀ : IsCompact s) (h₁ : IsOpen s) :
IsCompact (f ⁻¹' s) | hf.isCompact_preimage_of_isOpen h₁ h₀ | theorem | IsCompact.preimage_of_isOpen | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"IsCompact",
"IsOpen",
"IsSpectralMap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsSpectralMap.continuous {f : α → β} (hf : IsSpectralMap f) : Continuous f | hf.toContinuous | theorem | IsSpectralMap.continuous | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"Continuous",
"IsSpectralMap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isSpectralMap_id : IsSpectralMap (@id α) | ⟨continuous_id, fun _s _ => id⟩ | theorem | isSpectralMap_id | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"IsSpectralMap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsSpectralMap.comp {f : β → γ} {g : α → β} (hf : IsSpectralMap f) (hg : IsSpectralMap g) :
IsSpectralMap (f ∘ g) | ⟨hf.continuous.comp hg.continuous, fun _s hs₀ hs₁ =>
((hs₁.preimage_of_isOpen hf hs₀).preimage_of_isOpen hg) (hs₀.preimage hf.continuous)⟩ | theorem | IsSpectralMap.comp | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"IsSpectralMap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsProperMap.isSpectralMap {f : α → β} (hf : IsProperMap f) : IsSpectralMap f | ⟨hf.toContinuous, fun _ _ ↦ hf.isCompact_preimage⟩ | theorem | IsProperMap.isSpectralMap | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"IsProperMap",
"IsSpectralMap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SpectralMap (α β : Type*) [TopologicalSpace α] [TopologicalSpace β] where
/-- function between topological spaces -/
toFun : α → β
/-- proof that `toFun` is a spectral map -/
spectral' : IsSpectralMap toFun | structure | SpectralMap | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"IsSpectralMap",
"TopologicalSpace"
] | The type of spectral maps from `α` to `β`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SpectralMapClass (F α β : Type*) [TopologicalSpace α] [TopologicalSpace β]
[FunLike F α β] : Prop where
/-- statement that `F` is a type of spectral maps -/
map_spectral (f : F) : IsSpectralMap f | class | SpectralMapClass | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"FunLike",
"IsSpectralMap",
"TopologicalSpace"
] | `SpectralMapClass F α β` states that `F` is a type of spectral maps.
You should extend this class when you extend `SpectralMap`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toContinuousMap (f : SpectralMap α β) : ContinuousMap α β | ⟨_, f.spectral'.continuous⟩ | def | SpectralMap.toContinuousMap | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"ContinuousMap",
"SpectralMap",
"toContinuousMap"
] | Reinterpret a `SpectralMap` as a `ContinuousMap`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
instFunLike : FunLike (SpectralMap α β) α β | where
coe := SpectralMap.toFun
coe_injective' f g h := by cases f; cases g; congr | instance | SpectralMap.instFunLike | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"FunLike",
"SpectralMap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toFun_eq_coe {f : SpectralMap α β} : f.toFun = (f : α → β) | rfl | theorem | SpectralMap.toFun_eq_coe | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"SpectralMap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ext {f g : SpectralMap α β} (h : ∀ a, f a = g a) : f = g | DFunLike.ext f g h | theorem | SpectralMap.ext | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"DFunLike.ext",
"SpectralMap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
copy (f : SpectralMap α β) (f' : α → β) (h : f' = f) : SpectralMap α β | ⟨f', h.symm.subst f.spectral'⟩ | def | SpectralMap.copy | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"SpectralMap"
] | Copy of a `SpectralMap` with a new `toFun` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_copy (f : SpectralMap α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | theorem | SpectralMap.coe_copy | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"SpectralMap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
copy_eq (f : SpectralMap α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | DFunLike.ext' h | theorem | SpectralMap.copy_eq | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"DFunLike.ext'",
"SpectralMap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
id : SpectralMap α α | ⟨id, isSpectralMap_id⟩ | def | SpectralMap.id | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"SpectralMap"
] | `id` as a `SpectralMap`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_id : ⇑(SpectralMap.id α) = id | rfl | theorem | SpectralMap.coe_id | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"SpectralMap.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
id_apply (a : α) : SpectralMap.id α a = a | rfl | theorem | SpectralMap.id_apply | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"SpectralMap.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp (f : SpectralMap β γ) (g : SpectralMap α β) : SpectralMap α γ | ⟨f.toContinuousMap.comp g.toContinuousMap, f.spectral'.comp g.spectral'⟩ | def | SpectralMap.comp | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"SpectralMap"
] | Composition of `SpectralMap`s as a `SpectralMap`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_comp (f : SpectralMap β γ) (g : SpectralMap α β) : (f.comp g : α → γ) = f ∘ g | rfl | theorem | SpectralMap.coe_comp | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"SpectralMap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_apply (f : SpectralMap β γ) (g : SpectralMap α β) (a : α) : (f.comp g) a = f (g a) | rfl | theorem | SpectralMap.comp_apply | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"SpectralMap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_comp_continuousMap (f : SpectralMap β γ) (g : SpectralMap α β) :
f ∘ g = (f : ContinuousMap β γ) ∘ (g : ContinuousMap α β) | rfl | theorem | SpectralMap.coe_comp_continuousMap | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"ContinuousMap",
"SpectralMap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_comp_continuousMap' (f : SpectralMap β γ) (g : SpectralMap α β) :
(f.comp g : ContinuousMap α γ) = (f : ContinuousMap β γ).comp g | rfl | theorem | SpectralMap.coe_comp_continuousMap' | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"ContinuousMap",
"SpectralMap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_assoc (f : SpectralMap γ δ) (g : SpectralMap β γ) (h : SpectralMap α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | theorem | SpectralMap.comp_assoc | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"SpectralMap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_id (f : SpectralMap α β) : f.comp (SpectralMap.id α) = f | ext fun _a => rfl | theorem | SpectralMap.comp_id | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"SpectralMap",
"SpectralMap.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
id_comp (f : SpectralMap α β) : (SpectralMap.id β).comp f = f | ext fun _a => rfl | theorem | SpectralMap.id_comp | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"SpectralMap",
"SpectralMap.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cancel_right {g₁ g₂ : SpectralMap β γ} {f : SpectralMap α β} (hf : Surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨fun h => ext <| hf.forall.2 <| DFunLike.ext_iff.1 h,
fun a => of_eq (congrFun (congrArg comp a) f)⟩ | theorem | SpectralMap.cancel_right | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"SpectralMap",
"of_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cancel_left {g : SpectralMap β γ} {f₁ f₂ : SpectralMap α β} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨fun h => ext fun a => hg <| by rw [← comp_apply, h, comp_apply], congr_arg _⟩ | theorem | SpectralMap.cancel_left | Topology.Spectral | Mathlib/Topology/Spectral/Hom.lean | [] | [
"SpectralMap",
"f₁",
"f₂"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PrespectralSpace (X : Type*) [TopologicalSpace X] : Prop where
isTopologicalBasis : IsTopologicalBasis { U : Set X | IsOpen U ∧ IsCompact U } | class | PrespectralSpace | Topology.Spectral | Mathlib/Topology/Spectral/Prespectral.lean | [] | [
"IsCompact",
"IsOpen",
"Set",
"TopologicalSpace"
] | A space is prespectral if the lattice of compact opens forms a basis. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PrespectralSpace.of_isTopologicalBasis {B : Set (Set X)}
(basis : IsTopologicalBasis B) (isCompact_basis : ∀ U ∈ B, IsCompact U) :
PrespectralSpace X | where
isTopologicalBasis := basis.of_isOpen_of_subset (fun _ h ↦ h.1)
fun s hs ↦ ⟨basis.isOpen hs, isCompact_basis s hs⟩ | lemma | PrespectralSpace.of_isTopologicalBasis | Topology.Spectral | Mathlib/Topology/Spectral/Prespectral.lean | [] | [
"IsCompact",
"PrespectralSpace",
"Set"
] | A space is prespectral if it has a basis consisting of compact opens. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PrespectralSpace.of_isTopologicalBasis' {ι : Type*} {b : ι → Set X}
(basis : IsTopologicalBasis (Set.range b)) (isCompact_basis : ∀ i, IsCompact (b i)) :
PrespectralSpace X | .of_isTopologicalBasis basis (by simp_all) | lemma | PrespectralSpace.of_isTopologicalBasis' | Topology.Spectral | Mathlib/Topology/Spectral/Prespectral.lean | [] | [
"IsCompact",
"PrespectralSpace",
"Set",
"Set.range"
] | A space is prespectral if it has a basis consisting of compact opens.
This is the variant with an indexed basis instead. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PrespectralSpace.of_isOpenCover
{ι : Type*} {U : ι → Opens X} (hU : IsOpenCover U) [∀ i, PrespectralSpace (U i)] :
PrespectralSpace X | by
refine .of_isTopologicalBasis (hU.isTopologicalBasis fun i ↦ isTopologicalBasis) ?_
simp only [Set.mem_iUnion, Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp,
forall_comm (α := Set _), forall_apply_eq_imp_iff₂]
exact fun i V hV hV' ↦ hV'.image continuous_subtype_val | lemma | PrespectralSpace.of_isOpenCover | Topology.Spectral | Mathlib/Topology/Spectral/Prespectral.lean | [] | [
"PrespectralSpace",
"Set",
"Set.mem_iUnion",
"Set.mem_image",
"Set.mem_setOf_eq",
"continuous_subtype_val"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PrespectralSpace.of_isInducing [PrespectralSpace Y]
(f : X → Y) (hf : IsInducing f) (hf' : IsSpectralMap f) : PrespectralSpace X | .of_isTopologicalBasis (PrespectralSpace.isTopologicalBasis.isInducing hf) (by
simp only [Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp]
rintro _ U h₁ h₂ rfl
exact hf'.isCompact_preimage_of_isOpen h₁ h₂) | lemma | PrespectralSpace.of_isInducing | Topology.Spectral | Mathlib/Topology/Spectral/Prespectral.lean | [] | [
"IsSpectralMap",
"PrespectralSpace",
"Set.mem_image",
"Set.mem_setOf_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PrespectralSpace.of_isClosedEmbedding [PrespectralSpace Y]
(f : X → Y) (hf : IsClosedEmbedding f) : PrespectralSpace X | .of_isInducing f hf.isInducing hf.isProperMap.isSpectralMap | lemma | PrespectralSpace.of_isClosedEmbedding | Topology.Spectral | Mathlib/Topology/Spectral/Prespectral.lean | [] | [
"PrespectralSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Topology.IsOpenEmbedding.prespectralSpace [PrespectralSpace Y]
{f : X → Y} (hf : IsOpenEmbedding f) :
PrespectralSpace X | where
isTopologicalBasis := by
apply isTopologicalBasis_of_isOpen_of_nhds (fun U hU ↦ hU.1) <| fun x U hx hU ↦ ?_
obtain ⟨V, ⟨hoV, hcV⟩, hfx, hVf⟩ : ∃ V ∈ {V | IsOpen V ∧ IsCompact V}, f x ∈ V ∧ V ⊆ f '' U :=
(PrespectralSpace.isTopologicalBasis (X := Y)).isOpen_iff.mp
(hf.isOpen_iff_image_isOpe... | lemma | Topology.IsOpenEmbedding.prespectralSpace | Topology.Spectral | Mathlib/Topology/Spectral/Prespectral.lean | [] | [
"IsCompact",
"IsOpen",
"PrespectralSpace",
"Set.SurjOn.subset_range"
] | Let `f : X → Y` be an open embedding of topological spaces.
If `Y` is a prespectral space (i.e., the quasi-compact opens of `Y` form a basis),
then `X` is also a prespectral space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PrespectralSpace.sigma {ι : Type*} (X : ι → Type*) [∀ i, TopologicalSpace (X i)]
[∀ i, PrespectralSpace (X i)] : PrespectralSpace (Σ i, X i) | .of_isTopologicalBasis (IsTopologicalBasis.sigma fun i ↦ isTopologicalBasis) fun U hU ↦ by
simp_rw [Set.mem_iUnion] at hU
obtain ⟨i, V, hV, rfl⟩ := hU
exact hV.2.image continuous_sigmaMk | instance | PrespectralSpace.sigma | Topology.Spectral | Mathlib/Topology/Spectral/Prespectral.lean | [] | [
"PrespectralSpace",
"Set.mem_iUnion",
"TopologicalSpace",
"continuous_sigmaMk"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PrespectralSpace.isBasis_opens [PrespectralSpace X] :
TopologicalSpace.Opens.IsBasis { U : Opens X | IsCompact (U : Set X) } | by
dsimp only [TopologicalSpace.Opens.IsBasis]
convert! isTopologicalBasis (X := X)
ext s
exact ⟨fun ⟨V, hV, heq⟩ ↦ heq ▸ ⟨V.2, hV⟩, fun h ↦ ⟨⟨s, h.1⟩, h.2, rfl⟩⟩ | lemma | PrespectralSpace.isBasis_opens | Topology.Spectral | Mathlib/Topology/Spectral/Prespectral.lean | [] | [
"IsCompact",
"PrespectralSpace",
"Set",
"TopologicalSpace.Opens.IsBasis"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PrespectralSpace.opensEquiv [PrespectralSpace X] :
Opens X ≃o Order.Ideal (CompactOpens X) | where
toFun U := ⟨⟨{ V | (V : Set X) ⊆ U }, fun U₁ U₂ h₁ h₂ ↦ subset_trans (α := Set X) h₁ h₂⟩,
⟨⊥, by simp⟩, fun U₁ h₁ U₂ h₂ ↦ ⟨U₁ ⊔ U₂, by aesop, le_sup_left, le_sup_right⟩⟩
invFun I := ⨆ U ∈ I, U.toOpens
left_inv U := by
apply le_antisymm
· simp only [iSup_le_iff]
exact fun _ ↦ id
· intro... | def | PrespectralSpace.opensEquiv | Topology.Spectral | Mathlib/Topology/Spectral/Prespectral.lean | [] | [
"Order.Ideal",
"PrespectralSpace",
"Set",
"SetLike.coe_subset_coe",
"iSup_le_iff",
"le_antisymm",
"le_iSup₂",
"le_sup_left",
"le_trans",
"subset_trans"
] | In a prespectral space, the lattice of opens is determined by its lattice of compact opens. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsOpenMap.exists_opens_image_eq_of_prespectralSpace [PrespectralSpace X] {f : X → Y}
(hfc : Continuous f) (h : IsOpenMap f) {U : Set Y} (hs : U ⊆ Set.range f) (hU : IsOpen U)
(hc : IsCompact U) : ∃ (V : Opens X), IsCompact V.1 ∧ f '' V = U | by
obtain ⟨Us, hUs, heq⟩ := TopologicalSpace.Opens.isBasis_iff_cover.mp
(PrespectralSpace.isBasis_opens X) ⟨f ⁻¹' U, hU.preimage hfc⟩
obtain ⟨t, ht⟩ := by
refine hc.elim_finite_subcover (fun s : Us ↦ f '' s.1) (fun s ↦ h _ s.1.2) (fun x hx ↦ ?_)
obtain ⟨x, rfl⟩ := hs hx
obtain ⟨i, hi, hx⟩ := mem_sSu... | lemma | IsOpenMap.exists_opens_image_eq_of_prespectralSpace | Topology.Spectral | Mathlib/Topology/Spectral/Prespectral.lean | [] | [
"Continuous",
"IsCompact",
"IsOpen",
"IsOpenMap",
"PrespectralSpace",
"PrespectralSpace.isBasis_opens",
"Set",
"Set.iUnion_coe_set",
"Set.image_iUnion",
"Set.mem_iUnion",
"Set.range",
"subset_antisymm"
] | If `X` has a basis of compact opens and `f : X → S` is open, every
compact open of `S` is the image of a compact open of `X`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PrespectralSpace.exists_isCompact_and_isOpen_between [PrespectralSpace X] {K U : Set X}
(hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :
∃ (W : Set X), IsCompact W ∧ IsOpen W ∧ K ⊆ W ∧ W ⊆ U | by
refine hK.induction_on ⟨∅, by simp⟩ (fun s t hst ⟨W, Wc, Wo, hKW, hWU⟩ ↦ ?_) ?_ ?_
· use W, Wc, Wo, subset_trans hst hKW, hWU
· intro s t ⟨W₁, Wc₁, Wo₁, hKW₁, hWU₁⟩ ⟨W₂, Wc₂, Wo₂, hKW₂, hWU₂⟩
exact ⟨W₁ ∪ W₂, Wc₁.union Wc₂, Wo₁.union Wo₂, Set.union_subset_union hKW₁ hKW₂,
Set.union_subset hWU₁ hWU₂⟩
... | lemma | PrespectralSpace.exists_isCompact_and_isOpen_between | Topology.Spectral | Mathlib/Topology/Spectral/Prespectral.lean | [] | [
"IsCompact",
"IsOpen",
"PrespectralSpace",
"Set",
"Set.union_subset",
"Set.union_subset_union",
"subset_rfl",
"subset_trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PrespectralSpace.exists_isClosed_of_not_isPreirreducible [PrespectralSpace X] (Z : Set X)
(hZ : ¬ IsPreirreducible Z) :
∃ (A B : Set X), IsClosed A ∧ IsClosed B ∧ IsCompact Aᶜ ∧ IsCompact Bᶜ ∧
Z ⊆ A ∪ B ∧ (Z ∩ Aᶜ).Nonempty ∧ (Z ∩ Bᶜ).Nonempty | by
simp only [IsPreirreducible, not_forall] at hZ
rcases hZ with ⟨U₁, U₂, hU₁, hU₂, hU₁Z, hU₂Z, hU₁₂⟩
rw [Set.not_nonempty_iff_eq_empty, ← Set.subset_empty_iff] at hU₁₂
obtain ⟨x₁, hx₁⟩ : ∃ x₁ ∈ U₁, x₁ ∈ Z ∧ x₁ ∉ U₂ := by
obtain ⟨x, hx⟩ := hU₁Z
use x, hx.2, hx.1, fun h₂ ↦ hU₁₂ ⟨hx.1, hx.2, h₂⟩
obtain ... | lemma | PrespectralSpace.exists_isClosed_of_not_isPreirreducible | Topology.Spectral | Mathlib/Topology/Spectral/Prespectral.lean | [] | [
"IsClosed",
"IsCompact",
"IsPreirreducible",
"PrespectralSpace",
"Set",
"Set.mem_compl_iff",
"Set.mem_union",
"Set.not_nonempty_iff_eq_empty",
"Set.subset_empty_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformSpace : UniformSpace R | .ofFun (fun x y => abv (y - x)) (by simp) (fun x y => abv.map_sub y x)
(fun _ _ _ => (abv.sub_le _ _ _).trans_eq (add_comm _ _))
fun ε ε0 => ⟨ε / 2, half_pos ε0, fun _ h₁ _ h₂ => (add_lt_add h₁ h₂).trans_eq (add_halves ε)⟩ | def | AbsoluteValue.uniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbsoluteValue.lean | [] | [
"UniformSpace",
"add_halves",
"half_pos"
] | The uniform structure coming from an absolute value. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
hasBasis_uniformity :
𝓤[abv.uniformSpace].HasBasis ((0 : 𝕜) < ·) fun ε => { p : R × R | abv (p.2 - p.1) < ε } | UniformSpace.hasBasis_ofFun (exists_gt _) _ _ _ _ _ | theorem | AbsoluteValue.hasBasis_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbsoluteValue.lean | [] | [
"UniformSpace.hasBasis_ofFun"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
AbstractCompletion.{v, u} (α : Type u) [UniformSpace α] where
/-- The underlying space of the completion. -/
space : Type v
/-- A map from a space to its completion. -/
coe : α → space
/-- The completion carries a uniform structure. -/
uniformStruct : UniformSpace space
/-- The completion is complete. -/
... | structure | AbstractCompletion. | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"CompleteSpace",
"DenseRange",
"IsUniformInducing",
"T0Space",
"UniformSpace"
] | A completion of `α` is the data of a complete separated uniform space
and a map from `α` with dense range and inducing the original uniform structure on `α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofComplete [T0Space α] [CompleteSpace α] : AbstractCompletion α | mk α id inferInstance inferInstance inferInstance .id denseRange_id | def | AbstractCompletion.ofComplete | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"CompleteSpace",
"T0Space",
"denseRange_id"
] | If `α` is complete, then it is an abstract completion of itself. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
closure_range : closure (range ι) = univ | pkg.dense.closure_range | theorem | AbstractCompletion.closure_range | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isDenseInducing : IsDenseInducing ι | ⟨pkg.isUniformInducing.isInducing, pkg.dense⟩ | theorem | AbstractCompletion.isDenseInducing | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"IsDenseInducing"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_coe : UniformContinuous ι | IsUniformInducing.uniformContinuous pkg.isUniformInducing | theorem | AbstractCompletion.uniformContinuous_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"IsUniformInducing.uniformContinuous",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_coe : Continuous ι | pkg.uniformContinuous_coe.continuous | theorem | AbstractCompletion.continuous_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
induction_on {p : hatα → Prop} (a : hatα) (hp : IsClosed { a | p a }) (ih : ∀ a, p (ι a)) :
p a | isClosed_property pkg.dense hp ih a | theorem | AbstractCompletion.induction_on | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"IsClosed",
"isClosed_property"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
funext [TopologicalSpace β] [T2Space β] {f g : hatα → β} (hf : Continuous f)
(hg : Continuous g) (h : ∀ a, f (ι a) = g (ι a)) : f = g | funext fun a => pkg.induction_on a (isClosed_eq hf hg) h | theorem | AbstractCompletion.funext | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"Continuous",
"T2Space",
"TopologicalSpace",
"isClosed_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
extend (f : α → β) : hatα → β | def | AbstractCompletion.extend | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [] | Extension of maps to completions | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
extend_def (hf : UniformContinuous f) : pkg.extend f = pkg.isDenseInducing.extend f | if_pos hf | theorem | AbstractCompletion.extend_def | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inseparable_extend_coe (hf : UniformContinuous f) (x : α) :
Inseparable (pkg.extend f (ι x)) (f x) | by
rw [extend_def _ hf]
exact pkg.isDenseInducing.inseparable_extend hf.continuous.continuousAt | theorem | AbstractCompletion.inseparable_extend_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"Inseparable",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
extend_coe [T2Space β] (hf : UniformContinuous f) (a : α) : (pkg.extend f) (ι a) = f a | by
rw [pkg.extend_def hf]
exact pkg.isDenseInducing.extend_eq hf.continuous a | theorem | AbstractCompletion.extend_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"T2Space",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_extend : UniformContinuous (pkg.extend f) | by
by_cases hf : UniformContinuous f
· rw [pkg.extend_def hf]
exact uniformContinuous_uniformly_extend pkg.isUniformInducing pkg.dense hf
· unfold AbstractCompletion.extend
rw [if_neg hf]
exact uniformContinuous_of_const fun a b => by congr 1 | theorem | AbstractCompletion.uniformContinuous_extend | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"AbstractCompletion.extend",
"UniformContinuous",
"by_cases",
"uniformContinuous_of_const",
"uniformContinuous_uniformly_extend"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_extend : Continuous (pkg.extend f) | pkg.uniformContinuous_extend.continuous | theorem | AbstractCompletion.continuous_extend | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformInducing_extend (h : IsUniformInducing f) :
IsUniformInducing (pkg.extend f) | by
rw [extend_def _ h.uniformContinuous]
exact pkg.isDenseInducing.isUniformInducing_extend pkg.isUniformInducing h | lemma | AbstractCompletion.isUniformInducing_extend | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"IsUniformInducing"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
extend_unique (hf : UniformContinuous f) {g : hatα → β} (hg : UniformContinuous g)
(h : ∀ a : α, f a = g (ι a)) : pkg.extend f = g | by
apply pkg.funext pkg.continuous_extend hg.continuous
simpa only [pkg.extend_coe hf] using h | theorem | AbstractCompletion.extend_unique | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
extend_comp_coe {f : hatα → β} (hf : UniformContinuous f) : pkg.extend (f ∘ ι) = f | funext fun x =>
pkg.induction_on x (isClosed_eq pkg.continuous_extend hf.continuous) fun y =>
pkg.extend_coe (hf.comp <| pkg.uniformContinuous_coe) y | theorem | AbstractCompletion.extend_comp_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"UniformContinuous",
"isClosed_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map (f : α → β) : hatα → hatβ | pkg.extend (ι' ∘ f) | def | AbstractCompletion.map | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [] | Lifting maps to completions | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformContinuous_map : UniformContinuous (map f) | pkg.uniformContinuous_extend | theorem | AbstractCompletion.uniformContinuous_map | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_map : Continuous (map f) | pkg.continuous_extend | theorem | AbstractCompletion.continuous_map | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_coe (hf : UniformContinuous f) (a : α) : map f (ι a) = ι' (f a) | pkg.extend_coe (pkg'.uniformContinuous_coe.comp hf) a | theorem | AbstractCompletion.map_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_unique {f : α → β} {g : hatα → hatβ} (hg : UniformContinuous g)
(h : ∀ a, ι' (f a) = g (ι a)) : map f = g | pkg.funext (pkg.continuous_map _ _) hg.continuous <| by
intro a
change pkg.extend (ι' ∘ f) _ = _
simp_rw [Function.comp_def, h, ← comp_apply (f := g)]
rw [pkg.extend_coe (hg.comp pkg.uniformContinuous_coe)] | theorem | AbstractCompletion.map_unique | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_id : pkg.map pkg id = id | pkg.map_unique pkg uniformContinuous_id fun _ => rfl | theorem | AbstractCompletion.map_id | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"uniformContinuous_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
extend_map [CompleteSpace γ] [T0Space γ] {f : β → γ} {g : α → β}
(hf : UniformContinuous f) (hg : UniformContinuous g) :
pkg'.extend f ∘ map g = pkg.extend (f ∘ g) | pkg.funext (pkg'.continuous_extend.comp (pkg.continuous_map pkg' _)) pkg.continuous_extend
fun a => by
rw [pkg.extend_coe (hf.comp hg), comp_apply, pkg.map_coe pkg' hg, pkg'.extend_coe hf]
rfl | theorem | AbstractCompletion.extend_map | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"CompleteSpace",
"T0Space",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_comp {g : β → γ} {f : α → β} (hg : UniformContinuous g) (hf : UniformContinuous f) :
pkg'.map pkg'' g ∘ pkg.map pkg' f = pkg.map pkg'' (g ∘ f) | pkg.extend_map pkg' (pkg''.uniformContinuous_coe.comp hg) hf | theorem | AbstractCompletion.map_comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mapEquiv (e : α ≃ᵤ β) : hatα ≃ᵤ hatβ | where
toFun := pkg.map pkg' e
invFun := pkg'.map pkg e.symm
uniformContinuous_toFun := uniformContinuous_map ..
uniformContinuous_invFun := uniformContinuous_map ..
left_inv := Function.leftInverse_iff_comp.2 <| by
simp [map_comp _ _ _ e.symm.uniformContinuous e.uniformContinuous]
right_inv := Function.... | def | AbstractCompletion.mapEquiv | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [] | The uniform isomorphism between two completions of isomorphic uniform spaces. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mapEquiv_symm (e : α ≃ᵤ β) :
(pkg.mapEquiv pkg' e).symm = pkg'.mapEquiv pkg e.symm | rfl | theorem | AbstractCompletion.mapEquiv_symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mapEquiv_coe (e : α ≃ᵤ β) (a : α) : pkg.mapEquiv pkg' e (ι a) = ι' (e a) | pkg.map_coe pkg' e.uniformContinuous _ | theorem | AbstractCompletion.mapEquiv_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compare : pkg.space → pkg'.space | pkg.extend pkg'.coe | def | AbstractCompletion.compare | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [] | The comparison map between two completions of the same uniform space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformContinuous_compare : UniformContinuous (pkg.compare pkg') | pkg.uniformContinuous_extend | theorem | AbstractCompletion.uniformContinuous_compare | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compare_coe (a : α) : pkg.compare pkg' (pkg.coe a) = pkg'.coe a | pkg.extend_coe pkg'.uniformContinuous_coe a | theorem | AbstractCompletion.compare_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inverse_compare : pkg.compare pkg' ∘ pkg'.compare pkg = id | by
have uc := pkg.uniformContinuous_compare pkg'
have uc' := pkg'.uniformContinuous_compare pkg
apply pkg'.funext (uc.comp uc').continuous continuous_id
intro a
rw [comp_apply, pkg'.compare_coe pkg, pkg.compare_coe pkg']
rfl | theorem | AbstractCompletion.inverse_compare | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"continuous_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compareEquiv : pkg.space ≃ᵤ pkg'.space | where
toFun := pkg.compare pkg'
invFun := pkg'.compare pkg
left_inv := congr_fun (pkg'.inverse_compare pkg)
right_inv := congr_fun (pkg.inverse_compare pkg')
uniformContinuous_toFun := uniformContinuous_compare _ _
uniformContinuous_invFun := uniformContinuous_compare _ _ | def | AbstractCompletion.compareEquiv | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [] | The uniform bijection between two completions of the same uniform space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformContinuous_compareEquiv : UniformContinuous (pkg.compareEquiv pkg') | pkg.uniformContinuous_compare pkg' | theorem | AbstractCompletion.uniformContinuous_compareEquiv | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_compareEquiv_symm : UniformContinuous (pkg.compareEquiv pkg').symm | pkg'.uniformContinuous_compare pkg | theorem | AbstractCompletion.uniformContinuous_compareEquiv_symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"UniformContinuous",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compare_comp_eq_compare (γ : Type uγ) [TopologicalSpace γ]
[T3Space γ] {f : α → γ} (cont_f : Continuous f) :
letI | pkg.uniformStruct.toTopologicalSpace
letI := pkg'.uniformStruct.toTopologicalSpace
(∀ a : pkg.space,
Filter.Tendsto f (Filter.comap pkg.coe (𝓝 a)) (𝓝 ((pkg.isDenseInducing.extend f) a))) →
pkg.isDenseInducing.extend f ∘ pkg'.compare pkg = pkg'.isDenseInducing.extend f := by
let _ := pkg'.uniform... | theorem | AbstractCompletion.compare_comp_eq_compare | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"AbstractCompletion.isDenseInducing",
"Continuous",
"Continuous.comp",
"Filter.Tendsto",
"Filter.comap",
"IsDenseInducing.continuous_extend",
"IsDenseInducing.extend_eq",
"IsDenseInducing.extend_unique",
"T3Space",
"TopologicalSpace",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
prod : AbstractCompletion (α × β) | where
space := hatα × hatβ
coe p := ⟨ι p.1, ι' p.2⟩
uniformStruct := inferInstance
complete := inferInstance
separation := inferInstance
isUniformInducing := IsUniformInducing.prod pkg.isUniformInducing pkg'.isUniformInducing
dense := pkg.dense.prodMap pkg'.dense | def | AbstractCompletion.prod | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"IsUniformInducing.prod"
] | Products of completions | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
extend₂ (f : α → β → γ) : hatα → hatβ → γ | curry <| (pkg.prod pkg').extend (uncurry f) | def | AbstractCompletion.extend₂ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [] | Extend two variable map to completions. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
extension₂_coe_coe (hf : UniformContinuous <| uncurry f) (a : α) (b : β) :
pkg.extend₂ pkg' f (ι a) (ι' b) = f a b | show (pkg.prod pkg').extend (uncurry f) ((pkg.prod pkg').coe (a, b)) = uncurry f (a, b) from
(pkg.prod pkg').extend_coe hf _ | theorem | AbstractCompletion.extension₂_coe_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_extension₂ : UniformContinuous₂ (pkg.extend₂ pkg' f) | by
rw [uniformContinuous₂_def, AbstractCompletion.extend₂, uncurry_curry]
apply uniformContinuous_extend | theorem | AbstractCompletion.uniformContinuous_extension₂ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"AbstractCompletion.extend₂",
"UniformContinuous₂",
"uniformContinuous₂_def"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map₂ (f : α → β → γ) : hatα → hatβ → hatγ | pkg.extend₂ pkg' (pkg''.coe ∘₂ f) | def | AbstractCompletion.map₂ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [] | Lift two variable maps to completions. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformContinuous_map₂ (f : α → β → γ) : UniformContinuous₂ (pkg.map₂ pkg' pkg'' f) | AbstractCompletion.uniformContinuous_extension₂ pkg pkg' _ | theorem | AbstractCompletion.uniformContinuous_map₂ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"AbstractCompletion.uniformContinuous_extension₂",
"UniformContinuous₂"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_map₂ {δ} [TopologicalSpace δ] {f : α → β → γ} {a : δ → hatα} {b : δ → hatβ}
(ha : Continuous a) (hb : Continuous b) :
Continuous fun d : δ => pkg.map₂ pkg' pkg'' f (a d) (b d) | (pkg.uniformContinuous_map₂ pkg' pkg'' f).continuous.comp₂ ha hb | theorem | AbstractCompletion.continuous_map₂ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"Continuous",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map₂_coe_coe (a : α) (b : β) (f : α → β → γ) (hf : UniformContinuous₂ f) :
pkg.map₂ pkg' pkg'' f (ι a) (ι' b) = ι'' (f a b) | pkg.extension₂_coe_coe (f := pkg''.coe ∘₂ f) pkg' (pkg''.uniformContinuous_coe.comp hf) a b | theorem | AbstractCompletion.map₂_coe_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | [] | [
"UniformContinuous₂"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Equicontinuous.comap_uniformFun_eq [CompactSpace X] (F_eqcont : Equicontinuous F) :
(UniformFun.uniformSpace X α).comap F =
(Pi.uniformSpace _).comap F | by
-- The `≤` inequality is trivial
refine le_antisymm (UniformSpace.comap_mono UniformFun.uniformContinuous_toFun) ?_
-- A bit of rewriting to get a nice intermediate statement.
simp_rw [UniformSpace.comap, UniformSpace.le_def, uniformity_comap, Pi.uniformity,
Filter.comap_iInf, comap_comap, Function.comp_... | theorem | Equicontinuous.comap_uniformFun_eq | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"CompactSpace",
"CompactSpace.elim_nhds_subcover",
"Equicontinuous",
"Filter.comap_iInf",
"Pi.uniformSpace",
"Pi.uniformity",
"Set",
"SetRel.prodMk_mem_comp",
"SetRel.symm",
"UniformFun.gen",
"UniformFun.hasBasis_uniformity",
"UniformFun.uniformContinuous_toFun",
"UniformFun.uniformSpace",
... | Let `X` be a compact topological space, `α` a uniform space, and `F : ι → (X → α)` an
equicontinuous family. Then, the uniform structures of uniform convergence and pointwise
convergence induce the same uniform structure on `ι`.
In other words, pointwise convergence and uniform convergence coincide on an equicontinuou... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Equicontinuous.isUniformInducing_uniformFun_iff_pi [UniformSpace ι] [CompactSpace X]
(F_eqcont : Equicontinuous F) :
IsUniformInducing (UniformFun.ofFun ∘ F) ↔ IsUniformInducing F | by
rw [isUniformInducing_iff_uniformSpace, isUniformInducing_iff_uniformSpace,
← F_eqcont.comap_uniformFun_eq]
rfl | lemma | Equicontinuous.isUniformInducing_uniformFun_iff_pi | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"CompactSpace",
"Equicontinuous",
"IsUniformInducing",
"UniformFun.ofFun",
"UniformSpace",
"isUniformInducing_iff_uniformSpace"
] | Let `X` be a compact topological space, `α` a uniform space, and `F : ι → (X → α)` an
equicontinuous family. Then, the uniform structures of uniform convergence and pointwise
convergence induce the same uniform structure on `ι`.
In other words, pointwise convergence and uniform convergence coincide on an equicontinuou... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Equicontinuous.inducing_uniformFun_iff_pi [TopologicalSpace ι] [CompactSpace X]
(F_eqcont : Equicontinuous F) :
IsInducing (UniformFun.ofFun ∘ F) ↔ IsInducing F | by
rw [isInducing_iff, isInducing_iff]
change (_ = (UniformFun.uniformSpace X α |>.comap F |>.toTopologicalSpace)) ↔
(_ = (Pi.uniformSpace _ |>.comap F |>.toTopologicalSpace))
rw [F_eqcont.comap_uniformFun_eq] | lemma | Equicontinuous.inducing_uniformFun_iff_pi | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"CompactSpace",
"Equicontinuous",
"Pi.uniformSpace",
"TopologicalSpace",
"UniformFun.ofFun",
"UniformFun.uniformSpace",
"comap"
] | Let `X` be a compact topological space, `α` a uniform space, and `F : ι → (X → α)` an
equicontinuous family. Then, the topologies of uniform convergence and pointwise convergence induce
the same topology on `ι`.
In other words, pointwise convergence and uniform convergence coincide on an equicontinuous
subset of `X → ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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