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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coe_prod (K : PositiveCompacts α) (L : PositiveCompacts β) :
(K ×ˢ L : PositiveCompacts (α × β)) = (K : Set α) ×ˢ (L : Set β) | rfl | theorem | TopologicalSpace.PositiveCompacts.coe_prod | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CompactOpens (α : Type*) [TopologicalSpace α] extends Compacts α where
isOpen' : IsOpen carrier | structure | TopologicalSpace.CompactOpens | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"IsOpen",
"TopologicalSpace"
] | The type of compact open sets of a topological space. This is useful in non-Hausdorff contexts,
in particular spectral spaces. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Simps.coe (s : CompactOpens α) : Set α | s
initialize_simps_projections CompactOpens (carrier → coe, as_prefix coe, as_prefix toCompacts) | def | TopologicalSpace.CompactOpens.Simps.coe | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | See Note [custom simps projection]. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isCompact (s : CompactOpens α) : IsCompact (s : Set α) | s.isCompact' | theorem | TopologicalSpace.CompactOpens.isCompact | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"IsCompact",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen (s : CompactOpens α) : IsOpen (s : Set α) | s.isOpen' | theorem | TopologicalSpace.CompactOpens.isOpen | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toOpens (s : CompactOpens α) : Opens α | ⟨s, s.isOpen⟩ | def | TopologicalSpace.CompactOpens.toOpens | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | Reinterpret a compact open as an open. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toClopens [T2Space α] (s : CompactOpens α) : Clopens α | ⟨s, s.isCompact.isClosed, s.isOpen⟩ | def | TopologicalSpace.CompactOpens.toClopens | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"T2Space"
] | Reinterpret a compact open as a clopen. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ext {s t : CompactOpens α} (h : (s : Set α) = t) : s = t | SetLike.ext' h | theorem | TopologicalSpace.CompactOpens.ext | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set",
"SetLike.ext'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_sup (s t : CompactOpens α) : ↑(s ⊔ t) = (s ∪ t : Set α) | rfl | lemma | TopologicalSpace.CompactOpens.coe_sup | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_bot : ↑(⊥ : CompactOpens α) = (∅ : Set α) | rfl | lemma | TopologicalSpace.CompactOpens.coe_bot | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.CompactOpens.coe_finsetSup | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Finset",
"Finset.induction_on",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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⟩ | instance | TopologicalSpace.CompactOpens.instInf | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_inf (s t : CompactOpens α) : ↑(s ⊓ t) = (s ∩ t : Set α) | rfl | lemma | TopologicalSpace.CompactOpens.coe_inf | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instSemilatticeInf : SemilatticeInf (CompactOpens α) | fast_instance% SetLike.coe_injective.semilatticeInf _ .rfl .rfl coe_inf | instance | TopologicalSpace.CompactOpens.instSemilatticeInf | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"SemilatticeInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instSDiff : SDiff (CompactOpens α) | where
sdiff s t := ⟨⟨s \ t, s.isCompact.diff t.isOpen⟩, s.isOpen.sdiff t.isCompact.isClosed⟩ | instance | TopologicalSpace.CompactOpens.instSDiff | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_sdiff (s t : CompactOpens α) : ↑(s \ t) = (s \ t : Set α) | rfl | lemma | TopologicalSpace.CompactOpens.coe_sdiff | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instGeneralizedBooleanAlgebra : GeneralizedBooleanAlgebra (CompactOpens α) | fast_instance% SetLike.coe_injective.generalizedBooleanAlgebra _
.rfl .rfl coe_sup coe_inf coe_bot coe_sdiff | instance | TopologicalSpace.CompactOpens.instGeneralizedBooleanAlgebra | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"GeneralizedBooleanAlgebra"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instTop : Top (CompactOpens α) | where top := ⟨⊤, isOpen_univ⟩ | instance | TopologicalSpace.CompactOpens.instTop | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Top"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_top : ↑(⊤ : CompactOpens α) = (univ : Set α) | rfl | lemma | TopologicalSpace.CompactOpens.coe_top | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instBoundedOrder : BoundedOrder (CompactOpens α) | fast_instance% BoundedOrder.lift ((↑) : _ → Set α) (fun _ _ => id) coe_top coe_bot | instance | TopologicalSpace.CompactOpens.instBoundedOrder | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"BoundedOrder",
"BoundedOrder.lift",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instCompl : Compl (CompactOpens α) | where
compl s := ⟨⟨sᶜ, s.isOpen.isClosed_compl.isCompact⟩, s.isCompact.isClosed.isOpen_compl⟩ | instance | TopologicalSpace.CompactOpens.instCompl | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Compl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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⟩ | instance | TopologicalSpace.CompactOpens.instHImp | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"HImp",
"IsClosed.isCompact",
"himp_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_compl (s : CompactOpens α) : ↑sᶜ = (sᶜ : Set α) | rfl | lemma | TopologicalSpace.CompactOpens.coe_compl | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_himp (s t : CompactOpens α) : ↑(s ⇨ t) = (s ⇨ t : Set α) | rfl | lemma | TopologicalSpace.CompactOpens.coe_himp | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instBooleanAlgebra : BooleanAlgebra (CompactOpens α) | fast_instance% SetLike.coe_injective.booleanAlgebra _
.rfl .rfl coe_sup coe_inf coe_top coe_bot coe_compl coe_sdiff coe_himp | instance | TopologicalSpace.CompactOpens.instBooleanAlgebra | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"BooleanAlgebra"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map (f : α → β) (hf : Continuous f) (hf' : IsOpenMap f) (s : CompactOpens α) : CompactOpens β | ⟨s.toCompacts.map f hf, hf' _ s.isOpen⟩ | def | TopologicalSpace.CompactOpens.map | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous",
"IsOpenMap"
] | The image of a compact open under a continuous open map. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_map {f : α → β} (hf : Continuous f) (hf' : IsOpenMap f) (s : CompactOpens α) :
(s.map f hf hf' : Set β) = f '' s | rfl | theorem | TopologicalSpace.CompactOpens.coe_map | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous",
"IsOpenMap",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_id (K : CompactOpens α) : K.map id continuous_id IsOpenMap.id = K | CompactOpens.ext <| Set.image_id _ | theorem | TopologicalSpace.CompactOpens.map_id | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set.image_id",
"continuous_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.CompactOpens.map_comp | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous",
"IsOpenMap",
"Set.image_comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_prod (K : CompactOpens α) (L : CompactOpens β) :
(K ×ˢ L : CompactOpens (α × β)) = (K : Set α) ×ˢ (L : Set β) | rfl | theorem | TopologicalSpace.CompactOpens.coe_prod | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOpenCover {ι X : Type*} [TopologicalSpace X] (u : ι → Opens X) : Prop | iSup u = ⊤ | def | TopologicalSpace.IsOpenCover | Topology.Sets | Mathlib/Topology/Sets/OpenCover.lean | [] | [
"TopologicalSpace",
"iSup"
] | An indexed family of open sets whose union is `X`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mk (h : iSup u = ⊤) : IsOpenCover u | h | lemma | TopologicalSpace.IsOpenCover.mk | Topology.Sets | Mathlib/Topology/Sets/OpenCover.lean | [] | [
"iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.IsOpenCover.of_sets | Topology.Sets | Mathlib/Topology/Sets/OpenCover.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iSup_eq_top (hu : IsOpenCover u) : ⨆ i, u i = ⊤ | hu | lemma | TopologicalSpace.IsOpenCover.iSup_eq_top | Topology.Sets | Mathlib/Topology/Sets/OpenCover.lean | [] | [
"iSup_eq_top"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.IsOpenCover.iSup_set_eq_univ | Topology.Sets | Mathlib/Topology/Sets/OpenCover.lean | [] | [
"Set",
"SetLike.coe_set_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.IsOpenCover.comap | Topology.Sets | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_mem (hu : IsOpenCover u) (a : X) : ∃ i, a ∈ u i | by
simpa [← hu.iSup_set_eq_univ] using mem_univ a | lemma | TopologicalSpace.IsOpenCover.exists_mem | Topology.Sets | Mathlib/Topology/Sets/OpenCover.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.IsOpenCover.exists_mem_nhds | Topology.Sets | Mathlib/Topology/Sets/OpenCover.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iUnion_inter (hu : IsOpenCover u) (s : Set X) :
⋃ i, s ∩ u i = s | by
simp [← inter_iUnion, hu.iSup_set_eq_univ] | lemma | TopologicalSpace.IsOpenCover.iUnion_inter | Topology.Sets | Mathlib/Topology/Sets/OpenCover.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.IsOpenCover.isTopologicalBasis | Topology.Sets | Mathlib/Topology/Sets/OpenCover.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Opens.IsBasis.isOpenCover {S : Set (Opens X)} (hS : Opens.IsBasis S) :
IsOpenCover (fun U : S ↦ (U : Opens X)) | top_le_iff.mp (subset_trans hS.2.superset (by simp)) | lemma | TopologicalSpace.Opens.IsBasis.isOpenCover | Topology.Sets | Mathlib/Topology/Sets/OpenCover.lean | [] | [
"Set",
"subset_trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Opens.IsBasis.isOpenCover_mem_and_le {S : Set (Opens X)} (hS : Opens.IsBasis S)
{U : ι → Opens X} (hU : IsOpenCover U) :
IsOpenCover (fun V : { x : Opens X × ι // x.1 ∈ S ∧ x.1 ≤ U x.2 } ↦ V.1.1) | by
refine top_le_iff.mp fun x _ ↦ ?_
obtain ⟨i, hxi⟩ := hU.exists_mem x
obtain ⟨_, ⟨V, hV, rfl⟩, hxV, hVU⟩ := hS.exists_subset_of_mem_open hxi (U i).2
simp only [Opens.iSup_mk, Opens.carrier_eq_coe, Opens.mem_mk, Set.mem_iUnion, SetLike.mem_coe]
exact ⟨⟨(V, i), hV, hVU⟩, hxV⟩ | lemma | TopologicalSpace.Opens.IsBasis.isOpenCover_mem_and_le | Topology.Sets | Mathlib/Topology/Sets/OpenCover.lean | [] | [
"Set",
"Set.mem_iUnion",
"SetLike.mem_coe"
] | Given an open cover and a basis,
the set of basis elements contained in any of the covers is still a cover. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsPreirreducible.of_subset_iUnion {X ι : Type*} [TopologicalSpace X]
{U : ι → Opens X} (hn : Pairwise ((¬ Disjoint · ·) on U))
(h : ∀ i, IsPreirreducible ((U i) : Set X))
{s : Set X} (hs : IsOpen s) (hsU : s ⊆ ⋃ i, U i) :
IsPreirreducible s | by
rcases s.eq_empty_or_nonempty with he | hne
· rw [he]; exact isPreirreducible_empty
· choose x hx using hne
choose i hi using mem_iUnion.mp <| hsU hx
rcases exists_mem_irreducibleComponents_subset_of_isIrreducible (U i).carrier ⟨⟨x, hi⟩, h i⟩
with ⟨u, hu, hUu⟩
by_cases huniv : s ⊆ u
· exa... | theorem | IsPreirreducible.of_subset_iUnion | Topology.Sets | Mathlib/Topology/Sets/OpenCover.lean | [] | [
"Disjoint",
"IsOpen",
"IsPreirreducible",
"Pairwise",
"Set",
"TopologicalSpace",
"by_cases",
"exists_mem_irreducibleComponents_subset_of_isIrreducible",
"isClosed_of_mem_irreducibleComponents",
"isPreirreducible_empty"
] | (Pre)Irreducibility of an open set can be checked on a cover by opens
with pairwise non-empty intersections. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PreirreducibleSpace.of_isOpenCover {X ι : Type*} [TopologicalSpace X]
{U : ι → Opens X} (hn : Pairwise ((¬ Disjoint · ·) on U)) (hU : IsOpenCover U)
(h : ∀ i, PreirreducibleSpace (U i)) :
PreirreducibleSpace X | have h' (i : _) : IsPreirreducible (U i).carrier := IsPreirreducible.of_subtype
⟨IsPreirreducible.of_subset_iUnion hn h' isOpen_univ (by simpa using hU.iSup_set_eq_univ)⟩ | theorem | PreirreducibleSpace.of_isOpenCover | Topology.Sets | Mathlib/Topology/Sets/OpenCover.lean | [] | [
"Disjoint",
"IsPreirreducible",
"IsPreirreducible.of_subtype",
"Pairwise",
"PreirreducibleSpace",
"TopologicalSpace",
"isOpen_univ"
] | (Pre)Irreducibility can be checked on an open cover with pairwise non-empty intersections. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | TopologicalSpace.Opens | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"IsOpen",
"Set"
] | The type of open subsets of a topological space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instSecondCountableOpens [SecondCountableTopology α] (U : Opens α) :
SecondCountableTopology U | inferInstanceAs (SecondCountableTopology U.1) | instance | TopologicalSpace.Opens.instSecondCountableOpens | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"SecondCountableTopology"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
«forall» {p : Opens α → Prop} : (∀ U, p U) ↔ ∀ (U : Set α) (hU : IsOpen U), p ⟨U, hU⟩ | ⟨fun h _ _ => h _, fun h _ => h _ _⟩ | theorem | TopologicalSpace.Opens.«forall» | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
carrier_eq_coe (U : Opens α) : U.1 = ↑U | rfl | theorem | TopologicalSpace.Opens.carrier_eq_coe | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_mk {U : Set α} {hU : IsOpen U} : ↑(⟨U, hU⟩ : Opens α) = U | rfl | theorem | TopologicalSpace.Opens.coe_mk | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"IsOpen",
"Set"
] | the coercion `Opens α → Set α` applied to a pair is the same as taking the first component | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_mk {x : α} {U : Set α} {h : IsOpen U} : x ∈ mk U h ↔ x ∈ U | Iff.rfl | theorem | TopologicalSpace.Opens.mem_mk | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nonempty_coeSort {U : Opens α} : Nonempty U ↔ (U : Set α).Nonempty | Set.nonempty_coe_sort | theorem | TopologicalSpace.Opens.nonempty_coeSort | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set",
"Set.nonempty_coe_sort"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nonempty_coe {U : Opens α} : (U : Set α).Nonempty ↔ ∃ x, x ∈ U | Iff.rfl | theorem | TopologicalSpace.Opens.nonempty_coe | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ext {U V : Opens α} (h : (U : Set α) = V) : U = V | SetLike.coe_injective h | theorem | TopologicalSpace.Opens.ext | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set",
"SetLike.coe_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_inj {U V : Opens α} : (U : Set α) = V ↔ U = V | SetLike.ext'_iff.symm | theorem | TopologicalSpace.Opens.coe_inj | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inclusion {U V : Opens α} (h : U ≤ V) : U → V | Set.inclusion h | abbrev | TopologicalSpace.Opens.inclusion | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set.inclusion"
] | A version of `Set.inclusion` not requiring definitional abuse | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen (U : Opens α) : IsOpen (U : Set α) | U.is_open' | theorem | TopologicalSpace.Opens.isOpen | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mk_coe (U : Opens α) : mk (↑U) U.isOpen = U | rfl | theorem | TopologicalSpace.Opens.mk_coe | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Simps.coe (U : Opens α) : Set α | U
initialize_simps_projections Opens (carrier → coe, as_prefix coe) | def | TopologicalSpace.Opens.Simps.coe | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set"
] | See Note [custom simps projection]. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
interior (s : Set α) : Opens α | ⟨interior s, isOpen_interior⟩ | def | TopologicalSpace.Opens.interior | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set",
"interior"
] | The interior of a set, as an element of `Opens`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_interior {s : Set α} {x : α} : x ∈ Opens.interior s ↔ x ∈ _root_.interior s | .rfl | theorem | TopologicalSpace.Opens.mem_interior | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set",
"mem_interior"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.Opens.gc | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"GaloisConnection",
"Set",
"interior_maximal",
"le_trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.Opens.gi | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"GaloisCoinsertion",
"interior_subset",
"le_antisymm"
] | The Galois coinsertion between sets and opens. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | theorem | TopologicalSpace.Opens.mk_inf_mk | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"IsOpen",
"IsOpen.inter",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_inf (s t : Opens α) : (↑(s ⊓ t) : Set α) = ↑s ∩ ↑t | rfl | theorem | TopologicalSpace.Opens.coe_inf | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_inf {s t : Opens α} {x : α} : x ∈ s ⊓ t ↔ x ∈ s ∧ x ∈ t | Iff.rfl | lemma | TopologicalSpace.Opens.mem_inf | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_sup (s t : Opens α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t | rfl | theorem | TopologicalSpace.Opens.coe_sup | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_sup {s t : Opens α} {x : α} : x ∈ (s ⊔ t) ↔ x ∈ s ∨ x ∈ t | .rfl | theorem | TopologicalSpace.Opens.mem_sup | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_bot : ((⊥ : Opens α) : Set α) = ∅ | rfl | theorem | TopologicalSpace.Opens.coe_bot | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_bot {x : α} : x ∈ (⊥ : Opens α) ↔ False | Iff.rfl | lemma | TopologicalSpace.Opens.mem_bot | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mk_empty : (⟨∅, isOpen_empty⟩ : Opens α) = ⊥ | rfl | theorem | TopologicalSpace.Opens.mk_empty | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_eq_empty {U : Opens α} : (U : Set α) = ∅ ↔ U = ⊥ | SetLike.coe_injective.eq_iff' rfl | theorem | TopologicalSpace.Opens.coe_eq_empty | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_top (x : α) : x ∈ (⊤ : Opens α) | trivial | lemma | TopologicalSpace.Opens.mem_top | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_top : ((⊤ : Opens α) : Set α) = Set.univ | rfl | theorem | TopologicalSpace.Opens.coe_top | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set",
"Set.univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mk_univ : (⟨univ, isOpen_univ⟩ : Opens α) = ⊤ | rfl | theorem | TopologicalSpace.Opens.mk_univ | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_eq_univ {U : Opens α} : (U : Set α) = univ ↔ U = ⊤ | SetLike.coe_injective.eq_iff' rfl | theorem | TopologicalSpace.Opens.coe_eq_univ | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_sSup {S : Set (Opens α)} : (↑(sSup S) : Set α) = ⋃ i ∈ S, ↑i | rfl | theorem | TopologicalSpace.Opens.coe_sSup | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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)) _ _ | theorem | TopologicalSpace.Opens.coe_finset_sup | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Finset",
"Set",
"SupBotHom",
"map_finset_sup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.Opens.coe_finset_inf | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Finset",
"InfTopHom",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_disjoint {s t : Opens α} : Disjoint (s : Set α) t ↔ Disjoint s t | by
simp [disjoint_iff, ← SetLike.coe_set_eq] | lemma | TopologicalSpace.Opens.coe_disjoint | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Disjoint",
"Set",
"SetLike.coe_set_eq",
"disjoint_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_iSup {ι} (s : ι → Opens α) : ((⨆ i, s i : Opens α) : Set α) = ⋃ i, s i | by
simp [iSup] | theorem | TopologicalSpace.Opens.coe_iSup | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set",
"iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iSup_def {ι} (s : ι → Opens α) : ⨆ i, s i = ⟨⋃ i, s i, isOpen_iUnion fun i => (s i).2⟩ | ext <| coe_iSup s | theorem | TopologicalSpace.Opens.iSup_def | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"isOpen_iUnion"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 _ | theorem | TopologicalSpace.Opens.iSup_mk | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"IsOpen",
"Set",
"isOpen_iUnion"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_iSup {ι} {x : α} {s : ι → Opens α} : x ∈ iSup s ↔ ∃ i, x ∈ s i | by
rw [← SetLike.mem_coe]
simp | theorem | TopologicalSpace.Opens.mem_iSup | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"SetLike.mem_coe",
"iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.Opens.mem_sSup | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set",
"sSup_eq_iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.Opens.frameMinimalAxioms | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set.inter_iUnion₂"
] | Open sets in a topological space form a frame. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
instFrame : Frame (Opens α) | fast_instance% .ofMinimalAxioms frameMinimalAxioms | instance | TopologicalSpace.Opens.instFrame | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"ofMinimalAxioms"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frameHom : FrameHom (Opens α) (Set α) | where
toFun := (·)
map_inf' _ _ := rfl
map_top' := rfl
map_sSup' _ := by simp | def | TopologicalSpace.Opens.frameHom | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"FrameHom",
"Set"
] | The coercion from open sets to sets as a `FrameHom`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpenEmbedding' (U : Opens α) : IsOpenEmbedding (Subtype.val : U → α) | U.isOpen.isOpenEmbedding_subtypeVal | theorem | TopologicalSpace.Opens.isOpenEmbedding' | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.Opens.isOpenEmbedding_of_le | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set.inclusion",
"Set.range_inclusion",
"continuous_subtype_val"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.Opens.not_nonempty_iff_eq_bot | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set",
"Set.Nonempty",
"Set.not_nonempty_iff_eq_empty"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ne_bot_iff_nonempty (U : Opens α) : U ≠ ⊥ ↔ Set.Nonempty (U : Set α) | by
rw [Ne, ← not_nonempty_iff_eq_bot, not_not] | theorem | TopologicalSpace.Opens.ne_bot_iff_nonempty | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set",
"Set.Nonempty"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_bot_or_top [IndiscreteTopology α] (U : Opens α) :
U = ⊥ ∨ U = ⊤ | by
rw [← coe_eq_empty, ← coe_eq_univ, ← IndiscreteTopology.isOpen_iff]
exact U.2 | theorem | TopologicalSpace.Opens.eq_bot_or_top | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"IndiscreteTopology",
"IndiscreteTopology.isOpen_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsBasis (B : Set (Opens α)) : Prop | IsTopologicalBasis (((↑) : _ → Set α) '' B) | def | TopologicalSpace.Opens.IsBasis | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set"
] | A set of `opens α` is a basis if the set of corresponding sets is a topological basis. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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
· i... | theorem | TopologicalSpace.Opens.isBasis_iff_nbhd | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"IsOpen.mem_nhds",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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... | theorem | TopologicalSpace.Opens.isBasis_iff_cover | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set",
"iSup_and",
"iSup_image",
"le_sSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.Opens.IsBasis.isCompact_open_iff_eq_finite_iUnion | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"IsCompact",
"IsOpen",
"Set",
"Set.range",
"isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis"
] | 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 | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 ... | lemma | TopologicalSpace.Opens.IsBasis.exists_finite_of_isCompact | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Finset.finite_toSet",
"IsCompact",
"Set",
"le_antisymm",
"le_trans",
"sSup_le_sSup",
"subset_trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.Opens.IsBasis.le_iff | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"IsOpen",
"Set",
"Set.subset_def",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.Opens.isBasis_sigma | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set",
"Set.image_comp",
"Set.image_iUnion",
"TopologicalSpace",
"TopologicalSpace.IsTopologicalBasis.sigma",
"isOpenMap_sigmaMk"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | lemma | TopologicalSpace.Opens.IsBasis.of_isInducing | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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