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 ⌀ |
|---|---|---|---|---|---|---|
isClosed_union_iff (hst : SeparatedNhds s t) : IsClosed (s ∪ t) ↔ IsClosed s ∧ IsClosed t :=
⟨fun h ↦ ⟨hst.isClosed_left_of_isClosed_union h, hst.isClosed_right_of_isClosed_union h⟩,
fun ⟨h1, h2⟩ ↦ h1.union h2⟩ | lemma | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | isClosed_union_iff | null |
Closeds (α : Type*) [TopologicalSpace α] where
/-- the carrier set, i.e. the points in this set -/
carrier : Set α
isClosed' : IsClosed carrier | structure | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | Closeds | The type of closed subsets of a topological space. |
isClosed (s : Closeds α) : IsClosed (s : Set α) :=
s.isClosed'
@[deprecated (since := "2025-04-20")] alias closed := isClosed | theorem | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | isClosed | null |
Simps.coe (s : Closeds α) : Set α := s
initialize_simps_projections Closeds (carrier → coe, as_prefix coe)
@[simp] | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | Simps.coe | See Note [custom simps projection]. |
carrier_eq_coe (s : Closeds α) : s.carrier = (s : Set α) := rfl
@[ext] | lemma | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | carrier_eq_coe | null |
protected ext {s t : Closeds α} (h : (s : Set α) = t) : s = t :=
SetLike.ext' h
@[simp] | theorem | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | ext | null |
coe_mk (s : Set α) (h) : (mk s h : Set α) = s :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | coe_mk | null |
mem_mk {s : Set α} {hs : IsClosed s} {x : α} : x ∈ (⟨s, hs⟩ : Closeds α) ↔ x ∈ s :=
.rfl | lemma | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | mem_mk | null |
@[simps]
protected closure (s : Set α) : Closeds α :=
⟨closure s, isClosed_closure⟩
@[simp] | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | closure | The closure of a set, as an element of `TopologicalSpace.Closeds`. |
mem_closure {s : Set α} {x : α} : x ∈ Closeds.closure s ↔ x ∈ closure s := .rfl | theorem | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | mem_closure | null |
gc : GaloisConnection Closeds.closure ((↑) : Closeds α → Set α) := fun _ U =>
⟨subset_closure.trans, fun h => closure_minimal h U.isClosed⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | gc | null |
closure_le {s : Set α} {t : Closeds α} : .closure s ≤ t ↔ s ⊆ t :=
t.isClosed.closure_subset_iff | lemma | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | closure_le | null |
gi : GaloisInsertion (@Closeds.closure α _) (↑) where
choice s hs := ⟨s, closure_eq_iff_isClosed.1 <| hs.antisymm subset_closure⟩
gc := gc
le_l_u _ := subset_closure
choice_eq _s hs := SetLike.coe_injective <| subset_closure.antisymm hs | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | gi | The Galois insertion between sets and closeds. |
instCompleteLattice : CompleteLattice (Closeds α) :=
CompleteLattice.copy
(GaloisInsertion.liftCompleteLattice gi)
_ rfl
⟨univ, isClosed_univ⟩ rfl
⟨∅, isClosed_empty⟩ (SetLike.coe_injective closure_empty.symm)
(fun s t => ⟨s ∪ t, s.2.union t.2⟩)
(funext fun s => funext fun t => SetLike.coe_inj... | instance | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | instCompleteLattice | null |
coframeMinimalAxioms : Coframe.MinimalAxioms (Closeds α) where
iInf_sup_le_sup_sInf a s :=
(SetLike.coe_injective <| by simp only [coe_sup, coe_iInf, coe_sInf, Set.union_iInter₂]).le | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | coframeMinimalAxioms | The type of closed sets is inhabited, with default element the empty set. -/
instance : Inhabited (Closeds α) :=
⟨⊥⟩
@[simp, norm_cast]
theorem coe_sup (s t : Closeds α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t := by
rfl
@[simp, norm_cast]
theorem coe_inf (s t : Closeds α) : (↑(s ⊓ t) : Set α) = ↑s ∩ ↑t :=
rfl
@[simp, n... |
instCoframe : Coframe (Closeds α) := .ofMinimalAxioms coframeMinimalAxioms | instance | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | instCoframe | null |
@[simps]
singleton [T1Space α] (x : α) : Closeds α :=
⟨{x}, isClosed_singleton⟩
@[simp] lemma mem_singleton [T1Space α] {a b : α} : a ∈ singleton b ↔ a = b := Iff.rfl | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | singleton | The term of `TopologicalSpace.Closeds α` corresponding to a singleton. |
@[simps]
preimage (s : Closeds β) {f : α → β} (hf : Continuous f) : Closeds α :=
⟨f ⁻¹' s, s.isClosed.preimage hf⟩ | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | preimage | The preimage of a closed set under a continuous map. |
@[simps]
Closeds.compl (s : Closeds α) : Opens α :=
⟨sᶜ, s.2.isOpen_compl⟩ | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | Closeds.compl | The complement of a closed set as an open set. |
@[simps]
Opens.compl (s : Opens α) : Closeds α :=
⟨sᶜ, s.2.isClosed_compl⟩
nonrec theorem Closeds.compl_compl (s : Closeds α) : s.compl.compl = s :=
Closeds.ext (compl_compl (s : Set α))
nonrec theorem Opens.compl_compl (s : Opens α) : s.compl.compl = s :=
Opens.ext (compl_compl (s : Set α)) | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | Opens.compl | The complement of an open set as a closed set. |
Closeds.compl_bijective : Function.Bijective (@Closeds.compl α _) :=
Function.bijective_iff_has_inverse.mpr ⟨Opens.compl, Closeds.compl_compl, Opens.compl_compl⟩ | theorem | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | Closeds.compl_bijective | null |
Opens.compl_bijective : Function.Bijective (@Opens.compl α _) :=
Function.bijective_iff_has_inverse.mpr ⟨Closeds.compl, Opens.compl_compl, Closeds.compl_compl⟩
variable (α) | theorem | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | Opens.compl_bijective | null |
@[simps]
Closeds.complOrderIso : Closeds α ≃o (Opens α)ᵒᵈ where
toFun := OrderDual.toDual ∘ Closeds.compl
invFun := Opens.compl ∘ OrderDual.ofDual
left_inv s := by simp [Closeds.compl_compl]
right_inv s := by simp [Opens.compl_compl]
map_rel_iff' := (@OrderDual.toDual_le_toDual (Opens α)).trans compl_subset_c... | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | Closeds.complOrderIso | `TopologicalSpace.Closeds.compl` as an `OrderIso` to the order dual of
`TopologicalSpace.Opens α`. |
@[simps]
Opens.complOrderIso : Opens α ≃o (Closeds α)ᵒᵈ where
toFun := OrderDual.toDual ∘ Opens.compl
invFun := Closeds.compl ∘ OrderDual.ofDual
left_inv s := by simp [Opens.compl_compl]
right_inv s := by simp [Closeds.compl_compl]
map_rel_iff' := (@OrderDual.toDual_le_toDual (Closeds α)).trans compl_subset_c... | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | Opens.complOrderIso | `TopologicalSpace.Opens.compl` as an `OrderIso` to the order dual of
`TopologicalSpace.Closeds α`. |
Closeds.coe_eq_singleton_of_isAtom [T0Space α] {s : Closeds α} (hs : IsAtom s) :
∃ a, (s : Set α) = {a} := by
refine minimal_nonempty_closed_eq_singleton s.2 (coe_nonempty.2 hs.1) fun t hts ht ht' ↦ ?_
lift t to Closeds α using ht'
exact SetLike.coe_injective.eq_iff.2 <| (hs.le_iff_eq <| coe_nonempty.1 ht).1 ... | lemma | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | Closeds.coe_eq_singleton_of_isAtom | null |
Closeds.isAtom_iff [T1Space α] {s : Closeds α} :
IsAtom s ↔ ∃ x, s = Closeds.singleton x := by
simp [← Closeds.isAtom_coe, Set.isAtom_iff, SetLike.ext_iff, Set.ext_iff] | theorem | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | Closeds.isAtom_iff | in a `T1Space`, atoms of `TopologicalSpace.Closeds α` are precisely the
`TopologicalSpace.Closeds.singleton`s. |
Opens.isCoatom_iff [T1Space α] {s : Opens α} :
IsCoatom s ↔ ∃ x, s = (Closeds.singleton x).compl := by
rw [← s.compl_compl, ← isAtom_dual_iff_isCoatom]
change IsAtom (Closeds.complOrderIso α s.compl) ↔ _
simp only [(Closeds.complOrderIso α).isAtom_iff, Closeds.isAtom_iff,
Closeds.compl_bijective.injective... | theorem | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | Opens.isCoatom_iff | in a `T1Space`, coatoms of `TopologicalSpace.Opens α` are precisely complements of singletons:
`(TopologicalSpace.Closeds.singleton x).compl`. |
Clopens (α : Type*) [TopologicalSpace α] where
/-- the carrier set, i.e. the points in this set -/
carrier : Set α
isClopen' : IsClopen carrier | structure | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | Clopens | The type of clopen sets of a topological space. |
isClopen (s : Clopens α) : IsClopen (s : Set α) :=
s.isClopen' | theorem | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | isClopen | null |
isOpen (s : Clopens α) : IsOpen (s : Set α) := s.isClopen.isOpen | lemma | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | isOpen | null |
isClosed (s : Clopens α) : IsClosed (s : Set α) := s.isClopen.isClosed | lemma | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | isClosed | null |
Simps.coe (s : Clopens α) : Set α := s
initialize_simps_projections Clopens (carrier → coe, as_prefix coe) | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | Simps.coe | See Note [custom simps projection]. |
@[simps] toOpens (s : Clopens α) : Opens α := ⟨s, s.isOpen⟩ | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | toOpens | Reinterpret a clopen as an open. |
@[simps] toCloseds (s : Clopens α) : Closeds α := ⟨s, s.isClosed⟩
@[ext] | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | toCloseds | Reinterpret a clopen as a closed. |
protected ext {s t : Clopens α} (h : (s : Set α) = t) : s = t :=
SetLike.ext' h
@[simp] | theorem | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | ext | null |
coe_mk (s : Set α) (h) : (mk s h : Set α) = s :=
rfl
@[simp] lemma mem_mk {s : Set α} {x h} : x ∈ mk s h ↔ x ∈ s := .rfl | theorem | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | coe_mk | null |
@[simp, norm_cast] coe_sup (s t : Clopens α) : ↑(s ⊔ t) = (s ∪ t : Set α) := rfl
@[simp, norm_cast] lemma coe_inf (s t : Clopens α) : ↑(s ⊓ t) = (s ∩ t : Set α) := rfl
@[simp, norm_cast] lemma coe_top : (↑(⊤ : Clopens α) : Set α) = univ := rfl
@[simp, norm_cast] lemma coe_bot : (↑(⊥ : Clopens α) : Set α) = ∅ := rfl
@[s... | lemma | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | coe_sup | null |
@[simp]
protected mem_prod {s : Clopens α} {t : Clopens β} {x : α × β} :
x ∈ s ×ˢ t ↔ x.1 ∈ s ∧ x.2 ∈ t := .rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | mem_prod | null |
coe_finset_sup (s : Finset ι) (U : ι → Clopens α) :
(↑(s.sup U) : Set α) = ⋃ i ∈ s, U i := by
classical
induction s using Finset.induction_on with
| empty => simp
| insert _ _ _ IH => simp [IH]
@[simp, norm_cast] | lemma | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | coe_finset_sup | null |
coe_disjoint {s t : Clopens α} : Disjoint (s : Set α) t ↔ Disjoint s t := by
simp [disjoint_iff, ← SetLike.coe_set_eq] | lemma | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | coe_disjoint | null |
IrreducibleCloseds (α : Type*) [TopologicalSpace α] where
/-- the carrier set, i.e. the points in this set -/
carrier : Set α
is_irreducible' : IsIrreducible carrier
is_closed' : IsClosed carrier | structure | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | IrreducibleCloseds | The type of irreducible closed subsets of a topological space. |
isIrreducible (s : IrreducibleCloseds α) : IsIrreducible (s : Set α) := s.is_irreducible' | theorem | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | isIrreducible | null |
isClosed (s : IrreducibleCloseds α) : IsClosed (s : Set α) := s.is_closed' | theorem | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | isClosed | null |
Simps.coe (s : IrreducibleCloseds α) : Set α := s
initialize_simps_projections IrreducibleCloseds (carrier → coe, as_prefix coe)
@[ext] | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | Simps.coe | See Note [custom simps projection]. |
protected ext {s t : IrreducibleCloseds α} (h : (s : Set α) = t) : s = t :=
SetLike.ext' h
@[simp] | theorem | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | ext | null |
coe_mk (s : Set α) (h : IsIrreducible s) (h' : IsClosed s) : (mk s h h' : Set α) = s :=
rfl | theorem | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | coe_mk | null |
@[simps]
singleton [T1Space α] (x : α) : IrreducibleCloseds α :=
⟨{x}, isIrreducible_singleton, isClosed_singleton⟩
@[simp] lemma mem_singleton [T1Space α] {a b : α} : a ∈ singleton b ↔ a = b := Iff.rfl | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | singleton | The term of `TopologicalSpace.IrreducibleCloseds α` corresponding to a singleton. |
@[simps apply symm_apply]
equivSubtype : IrreducibleCloseds α ≃ { x : Set α // IsIrreducible x ∧ IsClosed x } where
toFun a := ⟨a.1, a.2, a.3⟩
invFun a := ⟨a.1, a.2.1, a.2.2⟩ | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | equivSubtype | The equivalence between `IrreducibleCloseds α` and `{x : Set α // IsIrreducible x ∧ IsClosed x }`. |
@[simps apply symm_apply]
equivSubtype' : IrreducibleCloseds α ≃ { x : Set α // IsClosed x ∧ IsIrreducible x } where
toFun a := ⟨a.1, a.3, a.2⟩
invFun a := ⟨a.1, a.2.2, a.2.1⟩
variable (α) in | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | equivSubtype' | The equivalence between `IrreducibleCloseds α` and `{x : Set α // IsClosed x ∧ IsIrreducible x }`. |
orderIsoSubtype : IrreducibleCloseds α ≃o { x : Set α // IsIrreducible x ∧ IsClosed x } :=
equivSubtype.toOrderIso (fun _ _ h ↦ h) (fun _ _ h ↦ h)
variable (α) in | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | orderIsoSubtype | The equivalence `IrreducibleCloseds α ≃ { x : Set α // IsIrreducible x ∧ IsClosed x }` is an
order isomorphism. |
orderIsoSubtype' : IrreducibleCloseds α ≃o { x : Set α // IsClosed x ∧ IsIrreducible x } :=
equivSubtype'.toOrderIso (fun _ _ h ↦ h) (fun _ _ h ↦ h) | def | Topology | [
"Mathlib.Topology.Sets.Opens",
"Mathlib.Topology.Clopen"
] | Mathlib/Topology/Sets/Closeds.lean | orderIsoSubtype' | The equivalence `IrreducibleCloseds α ≃ { x : Set α // IsClosed x ∧ IsIrreducible x }` is an
order isomorphism. |
IsCompactOpenCovered {S ι : Type*} {X : ι → Type*} (f : ∀ i, X i → S)
[∀ i, TopologicalSpace (X i)] (U : Set S) : Prop :=
∃ (s : Set ι) (_ : s.Finite) (V : ∀ i ∈ s, Opens (X i)),
(∀ (i : ι) (h : i ∈ s), IsCompact (V i h).1) ∧
⋃ (i : ι) (h : i ∈ s), (f i) '' (V i h) = U | def | Topology | [
"Mathlib.Data.Finite.Sigma",
"Mathlib.Topology.Spectral.Prespectral"
] | Mathlib/Topology/Sets/CompactOpenCovered.lean | IsCompactOpenCovered | A set `U` is compact-open covered by the family `fᵢ : X i → S`, if
`U` is the finite union of images of compact open sets in the `X i`. |
empty : IsCompactOpenCovered f ∅ :=
⟨∅, Set.finite_empty, fun _ _ ↦ ⟨∅, isOpen_empty⟩, fun _ _ ↦ isCompact_empty, by simp⟩ | lemma | Topology | [
"Mathlib.Data.Finite.Sigma",
"Mathlib.Topology.Spectral.Prespectral"
] | Mathlib/Topology/Sets/CompactOpenCovered.lean | empty | null |
iff_of_unique [Unique ι] :
IsCompactOpenCovered f U ↔ ∃ (V : Opens (X default)), IsCompact V.1 ∧ f default '' V.1 = U := by
refine ⟨fun ⟨s, hs, V, hc, hcov⟩ ↦ ?_, fun ⟨V, hc, h⟩ ↦ ?_⟩
· by_cases h : s = ∅
· aesop
· obtain rfl : s = {default} := by
rw [← Set.univ_unique, Subsingleton.eq_univ_of_n... | lemma | Topology | [
"Mathlib.Data.Finite.Sigma",
"Mathlib.Topology.Spectral.Prespectral"
] | Mathlib/Topology/Sets/CompactOpenCovered.lean | iff_of_unique | null |
id_iff_isOpen_and_isCompact [TopologicalSpace S] :
IsCompactOpenCovered (fun _ : Unit ↦ id) U ↔ IsOpen U ∧ IsCompact U := by
rw [iff_of_unique]
refine ⟨fun ⟨V, hV, heq⟩ ↦ ?_, fun ⟨ho, hc⟩ ↦ ⟨⟨U, ho⟩, hc, by simp⟩⟩
simp only [id_eq, Set.image_id', carrier_eq_coe, ← heq] at heq ⊢
exact ⟨V.2, hV⟩ | lemma | Topology | [
"Mathlib.Data.Finite.Sigma",
"Mathlib.Topology.Spectral.Prespectral"
] | Mathlib/Topology/Sets/CompactOpenCovered.lean | id_iff_isOpen_and_isCompact | null |
iff_isCompactOpenCovered_sigmaMk :
IsCompactOpenCovered f U ↔
IsCompactOpenCovered (fun (_ : Unit) (p : Σ i : ι, X i) ↦ f p.1 p.2) U := by
classical
rw [iff_of_unique (ι := Unit)]
refine ⟨fun ⟨s, hs, V, hc, hU⟩ ↦ ?_, fun ⟨V, hc, heq⟩ ↦ ?_⟩
· refine ⟨⟨s.sigma fun i ↦ if h : i ∈ s then V i h else ∅, isO... | lemma | Topology | [
"Mathlib.Data.Finite.Sigma",
"Mathlib.Topology.Spectral.Prespectral"
] | Mathlib/Topology/Sets/CompactOpenCovered.lean | iff_isCompactOpenCovered_sigmaMk | null |
of_iUnion_eq_of_finite (s : Set (Set S)) (hs : ⋃ t ∈ s, t = U) (hf : s.Finite)
(H : ∀ t ∈ s, IsCompactOpenCovered f t) : IsCompactOpenCovered f U := by
rw [iff_isCompactOpenCovered_sigmaMk, iff_of_unique]
have (t) (h : t ∈ s) : ∃ (V : Opens (Σ i, X i)),
IsCompact V.1 ∧ (fun p ↦ f p.fst p.snd) '' V.carrier... | lemma | Topology | [
"Mathlib.Data.Finite.Sigma",
"Mathlib.Topology.Spectral.Prespectral"
] | Mathlib/Topology/Sets/CompactOpenCovered.lean | of_iUnion_eq_of_finite | null |
exists_mem_of_isBasis {B : ∀ i, Set (Opens (X i))} (hB : ∀ i, IsBasis (B i))
(hBc : ∀ (i : ι), ∀ U ∈ B i, IsCompact U.1)
{U : Set S} (hU : IsCompactOpenCovered f U) :
∃ (n : ℕ) (a : Fin n → ι) (V : ∀ i, Opens (X (a i))),
(∀ i, V i ∈ B (a i)) ∧ ⋃ i, f (a i) '' V i = U := by
suffices h : ∃ (κ : Type _... | lemma | Topology | [
"Mathlib.Data.Finite.Sigma",
"Mathlib.Topology.Spectral.Prespectral"
] | Mathlib/Topology/Sets/CompactOpenCovered.lean | exists_mem_of_isBasis | If `U` is compact-open covered and the `X i` have a basis of compact opens,
`U` can be written as the union of images of elements of the basis. |
of_isOpenMap [TopologicalSpace S] [∀ i, PrespectralSpace (X i)]
(hfc : ∀ i, Continuous (f i)) (h : ∀ i, IsOpenMap (f i))
{U : Set S} (hs : ∀ x ∈ U, ∃ i y, f i y = x) (hU : IsOpen U) (hc : IsCompact U) :
IsCompactOpenCovered f U := by
rw [iff_isCompactOpenCovered_sigmaMk, iff_of_unique]
refine (isOpenMap... | lemma | Topology | [
"Mathlib.Data.Finite.Sigma",
"Mathlib.Topology.Spectral.Prespectral"
] | Mathlib/Topology/Sets/CompactOpenCovered.lean | of_isOpenMap | null |
Compacts (α : Type*) [TopologicalSpace α] where
/-- the carrier set, i.e. the points in this set -/
carrier : Set α
isCompact' : IsCompact carrier | structure | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | Compacts | The type of compact sets of a topological space. |
Simps.coe (s : Compacts α) : Set α := s
initialize_simps_projections Compacts (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 : Compacts α) : IsCompact (s : Set α) :=
s.isCompact' | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | isCompact | null |
@[ext]
protected ext {s t : Compacts 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 : Set α) (h) : (mk s h : Set α) = s :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | coe_mk | null |
carrier_eq_coe (s : Compacts α) : s.carrier = s :=
rfl | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | carrier_eq_coe | null |
protected map (f : α → β) (hf : Continuous f) (K : Compacts α) : Compacts β :=
⟨f '' K.1, K.2.image hf⟩
@[simp, norm_cast] | def | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | map | The type of compact sets is inhabited, with default element the empty set. -/
instance : Inhabited (Compacts α) := ⟨⊥⟩
@[simp]
theorem coe_sup (s t : Compacts α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t :=
rfl
@[simp]
theorem coe_inf [T2Space α] (s t : Compacts α) : (↑(s ⊓ t) : Set α) = ↑s ∩ ↑t :=
rfl
@[simp]
theorem coe_... |
coe_map {f : α → β} (hf : Continuous f) (s : Compacts α) : (s.map f 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 : Compacts α) : K.map id continuous_id = K :=
Compacts.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) (K : Compacts α) :
K.map (f ∘ g) (hf.comp hg) = (K.map g hg).map f hf :=
Compacts.ext <| Set.image_comp _ _ _ | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | map_comp | null |
@[simps]
protected equiv (f : α ≃ₜ β) : Compacts α ≃ Compacts β where
toFun := Compacts.map f f.continuous
invFun := Compacts.map _ f.symm.continuous
left_inv s := by
ext1
simp only [coe_map, ← image_comp, f.symm_comp_self, image_id]
right_inv s := by
ext1
simp only [coe_map, ← image_comp, f.sel... | def | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | equiv | A homeomorphism induces an equivalence on compact sets, by taking the image. |
equiv_refl : Compacts.equiv (Homeomorph.refl α) = Equiv.refl _ :=
Equiv.ext map_id
@[simp] | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | equiv_refl | null |
equiv_trans (f : α ≃ₜ β) (g : β ≃ₜ γ) :
Compacts.equiv (f.trans g) = (Compacts.equiv f).trans (Compacts.equiv g) :=
Equiv.ext <| map_comp g f g.continuous f.continuous
@[simp] | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | equiv_trans | null |
equiv_symm (f : α ≃ₜ β) : Compacts.equiv f.symm = (Compacts.equiv f).symm :=
rfl | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | equiv_symm | null |
coe_equiv_apply_eq_preimage (f : α ≃ₜ β) (K : Compacts α) :
(Compacts.equiv f K : Set β) = f.symm ⁻¹' (K : Set α) :=
f.toEquiv.image_eq_preimage K | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | coe_equiv_apply_eq_preimage | The image of a compact set under a homeomorphism can also be expressed as a preimage. |
protected prod (K : Compacts α) (L : Compacts β) : Compacts (α × β) where
carrier := K ×ˢ L
isCompact' := IsCompact.prod K.2 L.2
@[simp] | def | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | prod | The product of two `TopologicalSpace.Compacts`, as a `TopologicalSpace.Compacts` in the product
space. |
coe_prod (K : Compacts α) (L : Compacts 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 |
NonemptyCompacts (α : Type*) [TopologicalSpace α] extends Compacts α where
nonempty' : carrier.Nonempty | structure | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | NonemptyCompacts | The type of nonempty compact sets of a topological space. |
Simps.coe (s : NonemptyCompacts α) : Set α := s
initialize_simps_projections NonemptyCompacts (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 : NonemptyCompacts α) : IsCompact (s : Set α) :=
s.isCompact' | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | isCompact | null |
protected nonempty (s : NonemptyCompacts α) : (s : Set α).Nonempty :=
s.nonempty' | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | nonempty | null |
toCloseds [T2Space α] (s : NonemptyCompacts α) : Closeds α :=
⟨s, s.isCompact.isClosed⟩
@[ext] | def | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | toCloseds | Reinterpret a nonempty compact as a closed set. |
protected ext {s t : NonemptyCompacts 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 |
carrier_eq_coe (s : NonemptyCompacts α) : s.carrier = s :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | carrier_eq_coe | null |
coe_toCompacts (s : NonemptyCompacts α) : (s.toCompacts : Set α) = s := rfl | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | coe_toCompacts | null |
@[simp]
coe_sup (s t : NonemptyCompacts α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | coe_sup | null |
coe_top [CompactSpace α] [Nonempty α] : (↑(⊤ : NonemptyCompacts α) : Set α) = univ :=
rfl | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | coe_top | null |
protected prod (K : NonemptyCompacts α) (L : NonemptyCompacts β) : NonemptyCompacts (α × β) :=
{ K.toCompacts.prod L.toCompacts with nonempty' := K.nonempty.prod L.nonempty }
@[simp] | def | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | prod | In an inhabited space, the type of nonempty compact subsets is also inhabited, with
default element the singleton set containing the default element. -/
instance [Inhabited α] : Inhabited (NonemptyCompacts α) :=
⟨{ carrier := {default}
isCompact' := isCompact_singleton
nonempty' := singleton_nonempty _ }... |
coe_prod (K : NonemptyCompacts α) (L : NonemptyCompacts 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 |
PositiveCompacts (α : Type*) [TopologicalSpace α] extends Compacts α where
interior_nonempty' : (interior carrier).Nonempty | structure | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | PositiveCompacts | The type of compact sets with nonempty interior of a topological space.
See also `TopologicalSpace.Compacts` and `TopologicalSpace.NonemptyCompacts`. |
Simps.coe (s : PositiveCompacts α) : Set α := s
initialize_simps_projections PositiveCompacts (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 : PositiveCompacts α) : IsCompact (s : Set α) :=
s.isCompact' | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | isCompact | null |
interior_nonempty (s : PositiveCompacts α) : (interior (s : Set α)).Nonempty :=
s.interior_nonempty' | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | interior_nonempty | null |
protected nonempty (s : PositiveCompacts α) : (s : Set α).Nonempty :=
s.interior_nonempty.mono interior_subset | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | nonempty | null |
toNonemptyCompacts (s : PositiveCompacts α) : NonemptyCompacts α :=
⟨s.toCompacts, s.nonempty⟩
@[ext] | def | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | toNonemptyCompacts | Reinterpret a positive compact as a nonempty compact. |
protected ext {s t : PositiveCompacts 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 |
carrier_eq_coe (s : PositiveCompacts α) : s.carrier = s :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | carrier_eq_coe | null |
coe_toCompacts (s : PositiveCompacts α) : (s.toCompacts : Set α) = s :=
rfl | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | coe_toCompacts | null |
@[simp]
coe_sup (s t : PositiveCompacts α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.QuasiSeparated"
] | Mathlib/Topology/Sets/Compacts.lean | coe_sup | null |
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