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 ⌀ |
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coe_top [CompactSpace α] [Nonempty α] : (↑(⊤ : PositiveCompacts α) : Set α) = univ :=
rfl | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | coe_top | null |
protected map (f : α → β) (hf : Continuous f) (hf' : IsOpenMap f) (K : PositiveCompacts α) :
PositiveCompacts β :=
{ Compacts.map f hf K.toCompacts with
interior_nonempty' :=
(K.interior_nonempty'.image _).mono (hf'.image_interior_subset K.toCompacts) }
@[simp, norm_cast] | def | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | map | The image of a positive compact set under a continuous open map. |
coe_map {f : α → β} (hf : Continuous f) (hf' : IsOpenMap f) (s : PositiveCompacts α) :
(s.map f hf hf' : Set β) = f '' s :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | coe_map | null |
map_id (K : PositiveCompacts α) : K.map id continuous_id IsOpenMap.id = K :=
PositiveCompacts.ext <| Set.image_id _ | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | map_id | null |
map_comp (f : β → γ) (g : α → β) (hf : Continuous f) (hg : Continuous g) (hf' : IsOpenMap f)
(hg' : IsOpenMap g) (K : PositiveCompacts α) :
K.map (f ∘ g) (hf.comp hg) (hf'.comp hg') = (K.map g hg hg').map f hf hf' :=
PositiveCompacts.ext <| Set.image_comp _ _ _ | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | map_comp | null |
_root_.exists_positiveCompacts_subset [LocallyCompactSpace α] {U : Set α} (ho : IsOpen U)
(hn : U.Nonempty) : ∃ K : PositiveCompacts α, ↑K ⊆ U :=
let ⟨x, hx⟩ := hn
let ⟨K, hKc, hxK, hKU⟩ := exists_compact_subset ho hx
⟨⟨⟨K, hKc⟩, ⟨x, hxK⟩⟩, hKU⟩ | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | _root_.exists_positiveCompacts_subset | null |
_root_.IsOpen.exists_positiveCompacts_closure_subset [R1Space α] [LocallyCompactSpace α]
{U : Set α} (ho : IsOpen U) (hn : U.Nonempty) : ∃ K : PositiveCompacts α, closure ↑K ⊆ U :=
let ⟨K, hKU⟩ := exists_positiveCompacts_subset ho hn
⟨K, K.isCompact.closure_subset_of_isOpen ho hKU⟩ | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | _root_.IsOpen.exists_positiveCompacts_closure_subset | null |
nonempty' [WeaklyLocallyCompactSpace α] [Nonempty α] : Nonempty (PositiveCompacts α) := by
inhabit α
rcases exists_compact_mem_nhds (default : α) with ⟨K, hKc, hK⟩
exact ⟨⟨K, hKc⟩, _, mem_interior_iff_mem_nhds.2 hK⟩ | instance | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | nonempty' | In a nonempty locally compact space, there exists a compact set with nonempty interior. |
protected prod (K : PositiveCompacts α) (L : PositiveCompacts β) :
PositiveCompacts (α × β) where
toCompacts := K.toCompacts.prod L.toCompacts
interior_nonempty' := by
simp only [Compacts.carrier_eq_coe, Compacts.coe_prod, interior_prod_eq]
exact K.interior_nonempty.prod L.interior_nonempty
@[simp] | def | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | prod | The product of two `TopologicalSpace.PositiveCompacts`, as a `TopologicalSpace.PositiveCompacts`
in the product space. |
coe_prod (K : PositiveCompacts α) (L : PositiveCompacts β) :
(K.prod L : Set (α × β)) = (K : Set α) ×ˢ (L : Set β) :=
rfl | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | coe_prod | null |
CompactOpens (α : Type*) [TopologicalSpace α] extends Compacts α where
isOpen' : IsOpen carrier | structure | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | CompactOpens | The type of compact open sets of a topological space. This is useful in non-Hausdorff contexts,
in particular spectral spaces. |
Simps.coe (s : CompactOpens α) : Set α := s
initialize_simps_projections CompactOpens (carrier → coe, as_prefix coe) | def | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | Simps.coe | See Note [custom simps projection]. |
protected isCompact (s : CompactOpens α) : IsCompact (s : Set α) :=
s.isCompact' | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | isCompact | null |
protected isOpen (s : CompactOpens α) : IsOpen (s : Set α) :=
s.isOpen' | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | isOpen | null |
@[simps]
toOpens (s : CompactOpens α) : Opens α := ⟨s, s.isOpen⟩ | def | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | toOpens | Reinterpret a compact open as an open. |
@[simps]
toClopens [T2Space α] (s : CompactOpens α) : Clopens α :=
⟨s, s.isCompact.isClosed, s.isOpen⟩
@[ext] | def | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | toClopens | Reinterpret a compact open as a clopen. |
protected ext {s t : CompactOpens α} (h : (s : Set α) = t) : s = t :=
SetLike.ext' h
@[simp] | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | ext | null |
coe_mk (s : Compacts α) (h) : (mk s h : Set α) = s :=
rfl | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | coe_mk | null |
@[simp, norm_cast] coe_sup (s t : CompactOpens α) : ↑(s ⊔ t) = (s ∪ t : Set α) := rfl
@[simp, norm_cast] lemma coe_bot : ↑(⊥ : CompactOpens α) = (∅ : Set α) := rfl | lemma | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | coe_sup | null |
@[simp]
coe_finsetSup {ι : Type*} {f : ι → CompactOpens α} {s : Finset ι} :
(↑(s.sup f) : Set α) = ⋃ i ∈ s, f i := by
classical
induction s using Finset.induction_on <;> simp [*] | lemma | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | coe_finsetSup | null |
instInf : Min (CompactOpens α) where
min U V :=
⟨⟨U ∩ V, QuasiSeparatedSpace.inter_isCompact U.1.1 V.1.1 U.2 U.1.2 V.2 V.1.2⟩, U.2.inter V.2⟩
@[simp, norm_cast] lemma coe_inf (s t : CompactOpens α) : ↑(s ⊓ t) = (s ∩ t : Set α) := rfl | instance | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | instInf | null |
instSemilatticeInf : SemilatticeInf (CompactOpens α) :=
SetLike.coe_injective.semilatticeInf _ coe_inf | instance | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | instSemilatticeInf | null |
instSDiff : SDiff (CompactOpens α) where
sdiff s t := ⟨⟨s \ t, s.isCompact.diff t.isOpen⟩, s.isOpen.sdiff t.isCompact.isClosed⟩
@[simp, norm_cast] lemma coe_sdiff (s t : CompactOpens α) : ↑(s \ t) = (s \ t : Set α) := rfl | instance | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | instSDiff | null |
instGeneralizedBooleanAlgebra : GeneralizedBooleanAlgebra (CompactOpens α) :=
SetLike.coe_injective.generalizedBooleanAlgebra _ coe_sup coe_inf coe_bot coe_sdiff | instance | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | instGeneralizedBooleanAlgebra | null |
instTop : Top (CompactOpens α) where top := ⟨⊤, isOpen_univ⟩
@[simp, norm_cast] lemma coe_top : ↑(⊤ : CompactOpens α) = (univ : Set α) := rfl | instance | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | instTop | null |
instBoundedOrder : BoundedOrder (CompactOpens α) :=
BoundedOrder.lift ((↑) : _ → Set α) (fun _ _ => id) coe_top coe_bot | instance | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | instBoundedOrder | null |
instHasCompl : HasCompl (CompactOpens α) where
compl s := ⟨⟨sᶜ, s.isOpen.isClosed_compl.isCompact⟩, s.isCompact.isClosed.isOpen_compl⟩ | instance | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | instHasCompl | null |
instHImp : HImp (CompactOpens α) where
himp s t := ⟨⟨s ⇨ t, IsClosed.isCompact
(by simpa [himp_eq] using t.isCompact.isClosed.union s.isOpen.isClosed_compl)⟩,
by simpa [himp_eq] using t.isOpen.union s.isCompact.isClosed.isOpen_compl⟩
@[simp, norm_cast] lemma coe_compl (s : CompactOpens α) : ↑sᶜ = (sᶜ : Set α) := rfl
@[simp, norm_cast] lemma coe_himp (s t : CompactOpens α) : ↑(s ⇨ t) = (s ⇨ t : Set α) := rfl | instance | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | instHImp | null |
instBooleanAlgebra : BooleanAlgebra (CompactOpens α) :=
SetLike.coe_injective.booleanAlgebra _ coe_sup coe_inf coe_top coe_bot coe_compl coe_sdiff
coe_himp | instance | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | instBooleanAlgebra | null |
@[simps toCompacts]
map (f : α → β) (hf : Continuous f) (hf' : IsOpenMap f) (s : CompactOpens α) : CompactOpens β :=
⟨s.toCompacts.map f hf, hf' _ s.isOpen⟩
@[simp, norm_cast] | def | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | map | The image of a compact open under a continuous open map. |
coe_map {f : α → β} (hf : Continuous f) (hf' : IsOpenMap f) (s : CompactOpens α) :
(s.map f hf hf' : Set β) = f '' s :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | coe_map | null |
map_id (K : CompactOpens α) : K.map id continuous_id IsOpenMap.id = K :=
CompactOpens.ext <| Set.image_id _ | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | map_id | null |
map_comp (f : β → γ) (g : α → β) (hf : Continuous f) (hg : Continuous g) (hf' : IsOpenMap f)
(hg' : IsOpenMap g) (K : CompactOpens α) :
K.map (f ∘ g) (hf.comp hg) (hf'.comp hg') = (K.map g hg hg').map f hf hf' :=
CompactOpens.ext <| Set.image_comp _ _ _ | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | map_comp | null |
protected prod (K : CompactOpens α) (L : CompactOpens β) : CompactOpens (α × β) :=
{ K.toCompacts.prod L.toCompacts with isOpen' := K.isOpen.prod L.isOpen }
@[simp] | def | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | prod | The product of two `TopologicalSpace.CompactOpens`, as a `TopologicalSpace.CompactOpens` in the
product space. |
coe_prod (K : CompactOpens α) (L : CompactOpens β) :
(K.prod L : Set (α × β)) = (K : Set α) ×ˢ (L : Set β) :=
rfl | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | coe_prod | null |
IsOpenCover {ι X : Type*} [TopologicalSpace X] (u : ι → Opens X) : Prop :=
iSup u = ⊤
variable {ι κ X Y : Type*} [TopologicalSpace X] {u : ι → Opens X}
[TopologicalSpace Y] {v : κ → Opens Y} | def | Topology | [
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Sets/OpenCover.lean | IsOpenCover | An indexed family of open sets whose union is `X`. |
mk (h : iSup u = ⊤) : IsOpenCover u := h | lemma | Topology | [
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Sets/OpenCover.lean | mk | null |
of_sets {v : ι → Set X} (h_open : ∀ i, IsOpen (v i)) (h_iUnion : ⋃ i, v i = univ) :
IsOpenCover (fun i ↦ ⟨v i, h_open i⟩) := by
simp [IsOpenCover, h_iUnion] | lemma | Topology | [
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Sets/OpenCover.lean | of_sets | null |
iSup_eq_top (hu : IsOpenCover u) : ⨆ i, u i = ⊤ := hu | lemma | Topology | [
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Sets/OpenCover.lean | iSup_eq_top | null |
iSup_set_eq_univ (hu : IsOpenCover u) : ⋃ i, (u i : Set X) = univ := by
simpa [← SetLike.coe_set_eq] using hu.iSup_eq_top | lemma | Topology | [
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Sets/OpenCover.lean | iSup_set_eq_univ | null |
comap (hv : IsOpenCover v) (f : C(X, Y)) : IsOpenCover fun k ↦ (v k).comap f :=
by simp [IsOpenCover, ← preimage_iUnion, hv.iSup_set_eq_univ] | lemma | Topology | [
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Sets/OpenCover.lean | comap | Pullback of a covering of `Y` by a continuous map `X → Y`, giving a covering of `X` with the
same index type. |
exists_mem (hu : IsOpenCover u) (a : X) : ∃ i, a ∈ u i := by
simpa [← hu.iSup_set_eq_univ] using mem_univ a | lemma | Topology | [
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Sets/OpenCover.lean | exists_mem | null |
exists_mem_nhds (hu : IsOpenCover u) (a : X) : ∃ i, (u i : Set X) ∈ 𝓝 a :=
match hu.exists_mem a with | ⟨i, hi⟩ => ⟨i, (u i).isOpen.mem_nhds hi⟩ | lemma | Topology | [
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Sets/OpenCover.lean | exists_mem_nhds | null |
iUnion_inter (hu : IsOpenCover u) (s : Set X) :
⋃ i, s ∩ u i = s := by
simp [← inter_iUnion, hu.iSup_set_eq_univ] | lemma | Topology | [
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Sets/OpenCover.lean | iUnion_inter | null |
isTopologicalBasis (hu : IsOpenCover u)
{B : ∀ i, Set (Set (u i))} (hB : ∀ i, IsTopologicalBasis (B i)) :
IsTopologicalBasis (⋃ i, (Subtype.val '' ·) '' B i) :=
isTopologicalBasis_of_cover (fun i ↦ (u i).2) hu.iSup_set_eq_univ hB | lemma | Topology | [
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Sets/OpenCover.lean | isTopologicalBasis | null |
Opens where
/-- The underlying set of a bundled `TopologicalSpace.Opens` object. -/
carrier : Set α
/-- The `TopologicalSpace.Opens.carrier _` is an open set. -/
is_open' : IsOpen carrier | structure | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | Opens | The type of open subsets of a topological space. |
instSecondCountableOpens [SecondCountableTopology α] (U : Opens α) :
SecondCountableTopology U := inferInstanceAs (SecondCountableTopology U.1) | instance | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | instSecondCountableOpens | null |
@[simp] carrier_eq_coe (U : Opens α) : U.1 = ↑U := rfl | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | carrier_eq_coe | null |
@[simp]
coe_mk {U : Set α} {hU : IsOpen U} : ↑(⟨U, hU⟩ : Opens α) = U :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | coe_mk | the coercion `Opens α → Set α` applied to a pair is the same as taking the first component |
mem_mk {x : α} {U : Set α} {h : IsOpen U} : x ∈ mk U h ↔ x ∈ U := Iff.rfl | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | mem_mk | null |
protected nonempty_coeSort {U : Opens α} : Nonempty U ↔ (U : Set α).Nonempty :=
Set.nonempty_coe_sort | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | nonempty_coeSort | null |
protected nonempty_coe {U : Opens α} : (U : Set α).Nonempty ↔ ∃ x, x ∈ U :=
Iff.rfl
@[ext] -- TODO: replace with `∀ x, x ∈ U ↔ x ∈ V`? | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | nonempty_coe | null |
ext {U V : Opens α} (h : (U : Set α) = V) : U = V :=
SetLike.coe_injective h | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | ext | null |
coe_inj {U V : Opens α} : (U : Set α) = V ↔ U = V :=
SetLike.ext'_iff.symm | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | coe_inj | null |
inclusion {U V : Opens α} (h : U ≤ V) : U → V := Set.inclusion h | abbrev | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | inclusion | A version of `Set.inclusion` not requiring definitional abuse |
protected isOpen (U : Opens α) : IsOpen (U : Set α) :=
U.is_open'
@[simp] theorem mk_coe (U : Opens α) : mk (↑U) U.isOpen = U := rfl | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | isOpen | null |
Simps.coe (U : Opens α) : Set α := U
initialize_simps_projections Opens (carrier → coe, as_prefix coe) | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | Simps.coe | See Note [custom simps projection]. |
@[simps]
protected interior (s : Set α) : Opens α :=
⟨interior s, isOpen_interior⟩
@[simp] | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | interior | The interior of a set, as an element of `Opens`. |
mem_interior {s : Set α} {x : α} : x ∈ Opens.interior s ↔ x ∈ _root_.interior s := .rfl | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | mem_interior | null |
gc : GaloisConnection ((↑) : Opens α → Set α) Opens.interior := fun U _ =>
⟨fun h => interior_maximal h U.isOpen, fun h => le_trans h interior_subset⟩ | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | gc | null |
gi : GaloisCoinsertion (↑) (@Opens.interior α _) where
choice s hs := ⟨s, interior_eq_iff_isOpen.mp <| le_antisymm interior_subset hs⟩
gc := gc
u_l_le _ := interior_subset
choice_eq _s hs := le_antisymm hs interior_subset | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | gi | The Galois coinsertion between sets and opens. |
@[simp]
mk_inf_mk {U V : Set α} {hU : IsOpen U} {hV : IsOpen V} :
(⟨U, hU⟩ ⊓ ⟨V, hV⟩ : Opens α) = ⟨U ⊓ V, IsOpen.inter hU hV⟩ :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | mk_inf_mk | null |
coe_inf (s t : Opens α) : (↑(s ⊓ t) : Set α) = ↑s ∩ ↑t :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | coe_inf | null |
mem_inf {s t : Opens α} {x : α} : x ∈ s ⊓ t ↔ x ∈ s ∧ x ∈ t := Iff.rfl
@[simp, norm_cast] | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | mem_inf | null |
coe_sup (s t : Opens α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | coe_sup | null |
coe_bot : ((⊥ : Opens α) : Set α) = ∅ :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | coe_bot | null |
mem_bot {x : α} : x ∈ (⊥ : Opens α) ↔ False := Iff.rfl
@[simp] theorem mk_empty : (⟨∅, isOpen_empty⟩ : Opens α) = ⊥ := rfl
@[simp, norm_cast] | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | mem_bot | null |
coe_eq_empty {U : Opens α} : (U : Set α) = ∅ ↔ U = ⊥ :=
SetLike.coe_injective.eq_iff' rfl
@[simp] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | coe_eq_empty | null |
mem_top (x : α) : x ∈ (⊤ : Opens α) := trivial
@[simp, norm_cast] | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | mem_top | null |
coe_top : ((⊤ : Opens α) : Set α) = Set.univ :=
rfl
@[simp] theorem mk_univ : (⟨univ, isOpen_univ⟩ : Opens α) = ⊤ := rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | coe_top | null |
coe_eq_univ {U : Opens α} : (U : Set α) = univ ↔ U = ⊤ :=
SetLike.coe_injective.eq_iff' rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | coe_eq_univ | null |
coe_sSup {S : Set (Opens α)} : (↑(sSup S) : Set α) = ⋃ i ∈ S, ↑i :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | coe_sSup | null |
coe_finset_sup (f : ι → Opens α) (s : Finset ι) : (↑(s.sup f) : Set α) = s.sup ((↑) ∘ f) :=
map_finset_sup (⟨⟨(↑), coe_sup⟩, coe_bot⟩ : SupBotHom (Opens α) (Set α)) _ _
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | coe_finset_sup | null |
coe_finset_inf (f : ι → Opens α) (s : Finset ι) : (↑(s.inf f) : Set α) = s.inf ((↑) ∘ f) :=
map_finset_inf (⟨⟨(↑), coe_inf⟩, coe_top⟩ : InfTopHom (Opens α) (Set α)) _ _ | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | coe_finset_inf | null |
@[simp, norm_cast]
coe_iSup {ι} (s : ι → Opens α) : ((⨆ i, s i : Opens α) : Set α) = ⋃ i, s i := by
simp [iSup] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | coe_iSup | null |
iSup_def {ι} (s : ι → Opens α) : ⨆ i, s i = ⟨⋃ i, s i, isOpen_iUnion fun i => (s i).2⟩ :=
ext <| coe_iSup s
@[simp] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | iSup_def | null |
iSup_mk {ι} (s : ι → Set α) (h : ∀ i, IsOpen (s i)) :
(⨆ i, ⟨s i, h i⟩ : Opens α) = ⟨⋃ i, s i, isOpen_iUnion h⟩ :=
iSup_def _
@[simp] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | iSup_mk | null |
mem_iSup {ι} {x : α} {s : ι → Opens α} : x ∈ iSup s ↔ ∃ i, x ∈ s i := by
rw [← SetLike.mem_coe]
simp
@[simp] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | mem_iSup | null |
mem_sSup {Us : Set (Opens α)} {x : α} : x ∈ sSup Us ↔ ∃ u ∈ Us, x ∈ u := by
simp_rw [sSup_eq_iSup, mem_iSup, exists_prop] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | mem_sSup | null |
frameMinimalAxioms : Frame.MinimalAxioms (Opens α) where
inf_sSup_le_iSup_inf a s :=
(ext <| by simp only [coe_inf, coe_iSup, coe_sSup, Set.inter_iUnion₂]).le | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | frameMinimalAxioms | Open sets in a topological space form a frame. |
instFrame : Frame (Opens α) := .ofMinimalAxioms frameMinimalAxioms | instance | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | instFrame | null |
isOpenEmbedding' (U : Opens α) : IsOpenEmbedding (Subtype.val : U → α) :=
U.isOpen.isOpenEmbedding_subtypeVal | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | isOpenEmbedding' | null |
isOpenEmbedding_of_le {U V : Opens α} (i : U ≤ V) :
IsOpenEmbedding (Set.inclusion <| SetLike.coe_subset_coe.2 i) where
toIsEmbedding := .inclusion i
isOpen_range := by
rw [Set.range_inclusion i]
exact U.isOpen.preimage continuous_subtype_val | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | isOpenEmbedding_of_le | null |
not_nonempty_iff_eq_bot (U : Opens α) : ¬Set.Nonempty (U : Set α) ↔ U = ⊥ := by
rw [← coe_inj, coe_bot, ← Set.not_nonempty_iff_eq_empty] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | not_nonempty_iff_eq_bot | null |
ne_bot_iff_nonempty (U : Opens α) : U ≠ ⊥ ↔ Set.Nonempty (U : Set α) := by
rw [Ne, ← not_nonempty_iff_eq_bot, not_not] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | ne_bot_iff_nonempty | null |
eq_bot_or_top {α} [t : TopologicalSpace α] (h : t = ⊤) (U : Opens α) : U = ⊥ ∨ U = ⊤ := by
subst h; letI : TopologicalSpace α := ⊤
rw [← coe_eq_empty, ← coe_eq_univ, ← isOpen_top_iff]
exact U.2 | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | eq_bot_or_top | An open set in the indiscrete topology is either empty or the whole space. |
IsBasis (B : Set (Opens α)) : Prop :=
IsTopologicalBasis (((↑) : _ → Set α) '' B) | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | IsBasis | A set of `opens α` is a basis if the set of corresponding sets is a topological basis. |
isBasis_iff_nbhd {B : Set (Opens α)} :
IsBasis B ↔ ∀ {U : Opens α} {x}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U := by
constructor <;> intro h
· rintro ⟨sU, hU⟩ x hx
rcases h.mem_nhds_iff.mp (IsOpen.mem_nhds hU hx) with ⟨sV, ⟨⟨V, H₁, H₂⟩, hsV⟩⟩
refine ⟨V, H₁, ?_⟩
cases V
dsimp at H₂
subst H₂
exact hsV
· refine isTopologicalBasis_of_isOpen_of_nhds ?_ ?_
· rintro sU ⟨U, -, rfl⟩
exact U.2
· intro x sU hx hsU
rcases @h ⟨sU, hsU⟩ x hx with ⟨V, hV, H⟩
exact ⟨V, ⟨V, hV, rfl⟩, H⟩ | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | isBasis_iff_nbhd | null |
isBasis_iff_cover {B : Set (Opens α)} :
IsBasis B ↔ ∀ U : Opens α, ∃ Us, Us ⊆ B ∧ U = sSup Us := by
constructor
· intro hB U
refine ⟨{ V : Opens α | V ∈ B ∧ V ≤ U }, fun U hU => hU.left, ext ?_⟩
rw [coe_sSup, hB.open_eq_sUnion' U.isOpen]
simp_rw [sUnion_eq_biUnion, iUnion, mem_setOf_eq, iSup_and, iSup_image]
rfl
· intro h
rw [isBasis_iff_nbhd]
intro U x hx
rcases h U with ⟨Us, hUs, rfl⟩
rcases mem_sSup.1 hx with ⟨U, Us, xU⟩
exact ⟨U, hUs Us, xU, le_sSup Us⟩ | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | isBasis_iff_cover | null |
IsBasis.isCompact_open_iff_eq_finite_iUnion {ι : Type*} (b : ι → Opens α)
(hb : IsBasis (Set.range b)) (hb' : ∀ i, IsCompact (b i : Set α)) (U : Set α) :
IsCompact U ∧ IsOpen U ↔ ∃ s : Set ι, s.Finite ∧ U = ⋃ i ∈ s, b i := by
apply isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis fun i : ι => (b i).1
· convert (config := {transparency := .default}) hb
ext
simp
· exact hb' | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | IsBasis.isCompact_open_iff_eq_finite_iUnion | If `α` has a basis consisting of compact opens, then an open set in `α` is compact open iff
it is a finite union of some elements in the basis |
IsBasis.exists_finite_of_isCompact {B : Set (Opens α)} (hB : IsBasis B) {U : Opens α}
(hU : IsCompact U.1) : ∃ Us ⊆ B, Us.Finite ∧ U = sSup Us := by
classical
obtain ⟨Us', hsub, hsup⟩ := isBasis_iff_cover.mp hB U
obtain ⟨t, ht⟩ := hU.elim_finite_subcover (fun s : Us' ↦ s.1) (fun s ↦ s.1.2) (by simp [hsup])
refine ⟨Finset.image Subtype.val t, subset_trans (by simp) hsub, Finset.finite_toSet _, ?_⟩
exact le_antisymm (subset_trans ht (by simp)) (le_trans (sSup_le_sSup (by simp)) hsup.ge) | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | IsBasis.exists_finite_of_isCompact | null |
IsBasis.le_iff {α} {t₁ t₂ : TopologicalSpace α}
{Us : Set (Opens α)} (hUs : @IsBasis α t₂ Us) :
t₁ ≤ t₂ ↔ ∀ U ∈ Us, IsOpen[t₁] U := by
conv_lhs => rw [hUs.eq_generateFrom]
simp [Set.subset_def, le_generateFrom_iff_subset_isOpen] | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | IsBasis.le_iff | null |
isBasis_sigma {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)]
{B : ∀ i, Set (Opens (α i))} (hB : ∀ i, IsBasis (B i)) :
IsBasis (⋃ i : ι, (fun U ↦ ⟨Sigma.mk i '' U.1, isOpenMap_sigmaMk _ U.2⟩) '' B i) := by
convert TopologicalSpace.IsTopologicalBasis.sigma hB
simp only [IsBasis, Set.image_iUnion, ← Set.image_comp]
simp | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | isBasis_sigma | null |
IsBasis.of_isInducing {B : Set (Opens β)} (H : IsBasis B) {f : α → β} (h : IsInducing f) :
IsBasis { ⟨f ⁻¹' U, U.2.preimage h.continuous⟩ | U ∈ B } := by
simp only [IsBasis] at H ⊢
convert H.isInducing h
ext; simp
@[simp] | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | IsBasis.of_isInducing | null |
isCompactElement_iff (s : Opens α) :
CompleteLattice.IsCompactElement s ↔ IsCompact (s : Set α) := by
rw [isCompact_iff_finite_subcover, CompleteLattice.isCompactElement_iff]
refine ⟨?_, fun H ι U hU => ?_⟩
· introv H hU hU'
obtain ⟨t, ht⟩ := H ι (fun i => ⟨U i, hU i⟩) (by simpa)
refine ⟨t, Set.Subset.trans ht ?_⟩
rw [coe_finset_sup, Finset.sup_eq_iSup]
rfl
· obtain ⟨t, ht⟩ :=
H (fun i => U i) (fun i => (U i).isOpen) (by simpa using show (s : Set α) ⊆ ↑(iSup U) from hU)
refine ⟨t, Set.Subset.trans ht ?_⟩
simp only [Set.iUnion_subset_iff]
change ∀ i ∈ t, U i ≤ t.sup U
exact fun i => Finset.le_sup | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | isCompactElement_iff | null |
comap (f : C(α, β)) : FrameHom (Opens β) (Opens α) where
toFun s := ⟨f ⁻¹' s, s.2.preimage f.continuous⟩
map_sSup' s := ext <| by simp only [coe_sSup, preimage_iUnion, biUnion_image, coe_mk]
map_inf' _ _ := rfl
map_top' := rfl
@[simp] | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | comap | The preimage of an open set, as an open set. |
comap_id : comap (ContinuousMap.id α) = FrameHom.id _ :=
FrameHom.ext fun _ => ext rfl | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | comap_id | null |
comap_mono (f : C(α, β)) {s t : Opens β} (h : s ≤ t) : comap f s ≤ comap f t :=
OrderHomClass.mono (comap f) h
@[simp] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | comap_mono | null |
coe_comap (f : C(α, β)) (U : Opens β) : ↑(comap f U) = f ⁻¹' U :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | coe_comap | null |
mem_comap {f : C(α, β)} {U : Opens β} {x : α} : x ∈ comap f U ↔ f x ∈ U := .rfl | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Compactness.Bases",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Order.Copy"
] | Mathlib/Topology/Sets/Opens.lean | mem_comap | null |
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