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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isCompactElement_iff (s : Opens α) :
IsCompactElement s ↔ IsCompact (s : Set α) | by
rw [isCompact_iff_finite_subcover, CompleteLattice.isCompactElement_iff_exists_le_iSup_of_le_iSup]
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 ... | theorem | TopologicalSpace.Opens.isCompactElement_iff | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"Finset.le_sup",
"Finset.sup_eq_iSup",
"IsCompact",
"IsCompactElement",
"Set",
"Set.Subset.trans",
"Set.iUnion_subset_iff",
"iSup",
"isCompact_iff_finite_subcover"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | def | TopologicalSpace.Opens.comap | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"FrameHom",
"comap"
] | The preimage of an open set, as an open set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
comap_id : comap (ContinuousMap.id α) = FrameHom.id _ | FrameHom.ext fun _ => ext rfl | theorem | TopologicalSpace.Opens.comap_id | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"ContinuousMap.id",
"FrameHom.ext",
"FrameHom.id",
"comap",
"comap_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comap_mono (f : C(α, β)) {s t : Opens β} (h : s ≤ t) : comap f s ≤ comap f t | OrderHomClass.mono (comap f) h | theorem | TopologicalSpace.Opens.comap_mono | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"OrderHomClass.mono",
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_comap (f : C(α, β)) (U : Opens β) : ↑(comap f U) = f ⁻¹' U | rfl | theorem | TopologicalSpace.Opens.coe_comap | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_comap {f : C(α, β)} {U : Opens β} {x : α} : x ∈ comap f U ↔ f x ∈ U | .rfl | theorem | TopologicalSpace.Opens.mem_comap | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comap_comp (g : C(β, γ)) (f : C(α, β)) :
comap (g.comp f) = (comap f).comp (comap g) | rfl | theorem | TopologicalSpace.Opens.comap_comp | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"comap",
"comap_comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comap_comap (g : C(β, γ)) (f : C(α, β)) (U : Opens γ) :
comap f (comap g U) = comap (g.comp f) U | rfl | theorem | TopologicalSpace.Opens.comap_comap | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comap_injective [T0Space β] : Injective (comap : C(α, β) → FrameHom (Opens β) (Opens α)) | fun f g h =>
ContinuousMap.ext fun a =>
Inseparable.eq <|
inseparable_iff_forall_isOpen.2 fun s hs =>
have : comap f ⟨s, hs⟩ = comap g ⟨s, hs⟩ := DFunLike.congr_fun h ⟨_, hs⟩
show a ∈ f ⁻¹' s ↔ a ∈ g ⁻¹' s from Set.ext_iff.1 (coe_inj.2 this) a | theorem | TopologicalSpace.Opens.comap_injective | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"ContinuousMap.ext",
"DFunLike.congr_fun",
"FrameHom",
"Inseparable.eq",
"T0Space",
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Homeomorph.opensCongr (f : α ≃ₜ β) : Opens α ≃o Opens β | where
toFun := Opens.comap (f.symm : C(β, α))
invFun := Opens.comap (f : C(α, β))
left_inv _ := ext <| f.toEquiv.preimage_symm_preimage _
right_inv _ := ext <| f.toEquiv.symm_preimage_preimage _
map_rel_iff' := by
simp only [← SetLike.coe_subset_coe]; exact f.symm.surjective.preimage_subset_preimage_iff | def | Homeomorph.opensCongr | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"SetLike.coe_subset_coe"
] | A homeomorphism induces an order-preserving equivalence on open sets, by taking comaps. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_root_.Homeomorph.opensCongr_symm (f : α ≃ₜ β) : f.opensCongr.symm = f.symm.opensCongr | rfl | theorem | Homeomorph.opensCongr_symm | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
OpenNhdsOf (x : α) extends Opens α where
/-- The point `x` belongs to every `U : TopologicalSpace.OpenNhdsOf x`. -/
mem' : x ∈ carrier | structure | TopologicalSpace.OpenNhdsOf | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [] | The open neighborhoods of a point. See also `Opens` or `nhds`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toOpens_injective : Injective (toOpens : OpenNhdsOf x → Opens α) | | ⟨_, _⟩, ⟨_, _⟩, rfl => rfl | theorem | TopologicalSpace.OpenNhdsOf.toOpens_injective | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
canLiftSet : CanLift (Set α) (OpenNhdsOf x) (↑) fun s => IsOpen s ∧ x ∈ s | ⟨fun s hs => ⟨⟨⟨s, hs.1⟩, hs.2⟩, rfl⟩⟩ | instance | TopologicalSpace.OpenNhdsOf.canLiftSet | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"CanLift",
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem (U : OpenNhdsOf x) : x ∈ U | U.mem' | theorem | TopologicalSpace.OpenNhdsOf.mem | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen (U : OpenNhdsOf x) : IsOpen (U : Set α) | U.is_open' | theorem | TopologicalSpace.OpenNhdsOf.isOpen | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
basis_nhds : (𝓝 x).HasBasis (fun _ : OpenNhdsOf x => True) (↑) | (nhds_basis_opens x).to_hasBasis (fun U hU => ⟨⟨⟨U, hU.2⟩, hU.1⟩, trivial, Subset.rfl⟩) fun U _ =>
⟨U, ⟨⟨U.mem, U.isOpen⟩, Subset.rfl⟩⟩ | theorem | TopologicalSpace.OpenNhdsOf.basis_nhds | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comap (f : C(α, β)) (x : α) : LatticeHom (OpenNhdsOf (f x)) (OpenNhdsOf x) | where
toFun U := ⟨Opens.comap f U.1, U.mem⟩
map_sup' _ _ := rfl
map_inf' _ _ := rfl | def | TopologicalSpace.OpenNhdsOf.comap | Topology.Sets | Mathlib/Topology/Sets/Opens.lean | [] | [
"LatticeHom",
"comap"
] | Preimage of an open neighborhood of `f x` under a continuous map `f` as a `LatticeHom`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ClopenUpperSet (α : Type*) [TopologicalSpace α] [LE α] extends Clopens α where
upper' : IsUpperSet carrier | structure | ClopenUpperSet | Topology.Sets | Mathlib/Topology/Sets/Order.lean | [] | [
"IsUpperSet",
"TopologicalSpace"
] | The type of clopen upper sets of a topological space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Simps.coe (s : ClopenUpperSet α) : Set α | s
initialize_simps_projections ClopenUpperSet (carrier → coe, as_prefix coe) | def | ClopenUpperSet.Simps.coe | Topology.Sets | Mathlib/Topology/Sets/Order.lean | [] | [
"ClopenUpperSet",
"Set"
] | See Note [custom simps projection]. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
upper (s : ClopenUpperSet α) : IsUpperSet (s : Set α) | s.upper' | theorem | ClopenUpperSet.upper | Topology.Sets | Mathlib/Topology/Sets/Order.lean | [] | [
"ClopenUpperSet",
"IsUpperSet",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClopen (s : ClopenUpperSet α) : IsClopen (s : Set α) | s.isClopen' | theorem | ClopenUpperSet.isClopen | Topology.Sets | Mathlib/Topology/Sets/Order.lean | [] | [
"ClopenUpperSet",
"IsClopen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toUpperSet (s : ClopenUpperSet α) : UpperSet α | ⟨s, s.upper⟩ | def | ClopenUpperSet.toUpperSet | Topology.Sets | Mathlib/Topology/Sets/Order.lean | [] | [
"ClopenUpperSet",
"UpperSet"
] | Reinterpret an upper clopen as an upper set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ext {s t : ClopenUpperSet α} (h : (s : Set α) = t) : s = t | SetLike.ext' h | theorem | ClopenUpperSet.ext | Topology.Sets | Mathlib/Topology/Sets/Order.lean | [] | [
"ClopenUpperSet",
"Set",
"SetLike.ext'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_mk (s : Clopens α) (h) : (mk s h : Set α) = s | rfl | theorem | ClopenUpperSet.coe_mk | Topology.Sets | Mathlib/Topology/Sets/Order.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_sup (s t : ClopenUpperSet α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t | rfl | theorem | ClopenUpperSet.coe_sup | Topology.Sets | Mathlib/Topology/Sets/Order.lean | [] | [
"ClopenUpperSet",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_inf (s t : ClopenUpperSet α) : (↑(s ⊓ t) : Set α) = ↑s ∩ ↑t | rfl | theorem | ClopenUpperSet.coe_inf | Topology.Sets | Mathlib/Topology/Sets/Order.lean | [] | [
"ClopenUpperSet",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_top : (↑(⊤ : ClopenUpperSet α) : Set α) = univ | rfl | theorem | ClopenUpperSet.coe_top | Topology.Sets | Mathlib/Topology/Sets/Order.lean | [] | [
"ClopenUpperSet",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_bot : (↑(⊥ : ClopenUpperSet α) : Set α) = ∅ | rfl | theorem | ClopenUpperSet.coe_bot | Topology.Sets | Mathlib/Topology/Sets/Order.lean | [] | [
"ClopenUpperSet",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
vietoris : TopologicalSpace (Set α) | .generateFrom <| powerset '' {U | IsOpen U} ∪ (fun V => {s | (s ∩ V).Nonempty}) '' {V | IsOpen V} | def | TopologicalSpace.vietoris | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsOpen",
"Set",
"TopologicalSpace"
] | The Vietoris topology on the powerset of a topological space, generated by sets of the form
`{A | A ⊆ U}` and `{A | A ∩ U ≠ ∅}`, where `U` is an open subset of the underlying space. Used for
defining the topologies on `Compacts` and `NonemptyCompacts`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_root_.IsOpen.powerset_vietoris {U : Set α} (h : IsOpen U) :
IsOpen U.powerset | isOpen_generateFrom_of_mem <| .inl ⟨U, h, rfl⟩ | theorem | IsOpen.powerset_vietoris | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsOpen",
"Set"
] | When `Set` is equipped with the Vietoris topology, the powerset of an open set is open. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_inter_nonempty_of_isOpen {U : Set α} (h : IsOpen U) :
IsOpen {s | (s ∩ U).Nonempty} | isOpen_generateFrom_of_mem <| .inr ⟨U, h, rfl⟩ | theorem | TopologicalSpace.vietoris.isOpen_inter_nonempty_of_isOpen | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.IsClosed.powerset_vietoris {F : Set α} (h : IsClosed F) :
IsClosed F.powerset | by
simp_rw [powerset, ← isOpen_compl_iff, compl_setOf, ← inter_compl_nonempty_iff]
exact isOpen_inter_nonempty_of_isOpen h.isOpen_compl | theorem | IsClosed.powerset_vietoris | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsClosed",
"Set",
"isOpen_compl_iff"
] | When `Set` is equipped with the Vietoris topology, the powerset of a closed set is closed. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isClosed_inter_nonempty_of_isClosed {F : Set α} (h : IsClosed F) :
IsClosed {s | (s ∩ F).Nonempty} | by
simp_rw +singlePass [← compl_compl F, inter_compl_nonempty_iff, ← compl_setOf]
exact h.isOpen_compl.powerset_vietoris.isClosed_compl | theorem | TopologicalSpace.vietoris.isClosed_inter_nonempty_of_isClosed | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsClosed",
"Set",
"compl_compl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClopen_singleton_empty : IsClopen {(∅ : Set α)} | by
rw [← powerset_empty]
exact ⟨isClosed_empty.powerset_vietoris, isOpen_empty.powerset_vietoris⟩ | theorem | TopologicalSpace.vietoris.isClopen_singleton_empty | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsClopen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isTopologicalBasis :
IsTopologicalBasis <|
(fun u => {s ⊆ ⋃₀ u | ∀ U ∈ u, (s ∩ U).Nonempty}) ''
{u : Set (Set α) | u.Finite ∧ (∀ U ∈ u, IsOpen U)} | by
refine isTopologicalBasis_of_subbasis rfl |>.isTopologicalBasis_of_exists_subset ?_ ?_ <;>
rw [forall_mem_image]
· intro u ⟨hu₁, hu₂⟩
simp_rw [setOf_and, setOf_forall]
exact (isOpen_sUnion hu₂).powerset_vietoris.inter <|
hu₁.isOpen_biInter fun U hU => isOpen_inter_nonempty_of_isOpen (hu₂ U hU)
... | theorem | TopologicalSpace.vietoris.isTopologicalBasis | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsOpen",
"Set",
"isOpen_sUnion"
] | The Vietoris topology has a basis consisting of sets of the form
`{s | s ⊆ U₁ ∪ … ∪ Uₙ, s ∩ U₁ ≠ ∅, …, s ∩ Uₙ ≠ ∅}`, where `U₁, …, Uₙ` are open sets. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_root_.TopologicalSpace.IsTopologicalBasis.vietoris
{B : Set (Set α)} (hB : IsTopologicalBasis B) :
IsTopologicalBasis <|
(fun p => {s ⊆ p.1 | ∀ U ∈ p.2, (s ∩ U).Nonempty}) ''
{p : Set α × Set (Set α) | IsOpen p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ (∀ U ∈ p.2, U ⊆ p.1)} | by
refine isTopologicalBasis.isTopologicalBasis_of_exists_subset ?_ ?_ <;>
rw [forall_mem_image]
· intro ⟨V, u⟩ ⟨hV, hu, huB, _⟩
simp_rw [setOf_and, setOf_forall]
exact hV.powerset_vietoris.inter <|
hu.isOpen_biInter fun V hV => isOpen_inter_nonempty_of_isOpen <| hB.isOpen <| huB hV
· intro u ⟨h... | theorem | TopologicalSpace.IsTopologicalBasis.vietoris | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsOpen",
"Set"
] | Given a basis `B` on the underlying topological space, the Vietoris topology has a basis
consisting of sets of the form `{s | s ⊆ V, s ∩ U₁ ≠ ∅, …, s ∩ Uₙ ≠ ∅}`, where `U₁, …, Uₙ ∈ B` are
subsets of some open set `V`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
closure_finite_subsets (s : Set α) :
closure {t | t.Finite ∧ t ⊆ s} = (closure s).powerset | by
refine subset_antisymm ?_ (fun K hKs => ?_)
· rw [isClosed_closure.powerset_vietoris.closure_subset_iff]
exact fun K ⟨_, h⟩ => h.trans subset_closure
· rw [isTopologicalBasis.mem_closure_iff, forall_mem_image]
rintro u ⟨hu₁, hu₂⟩ ⟨ht₁, ht₂⟩
choose x hxU hxs using fun U : u => show (↑U ∩ s).Nonempty... | theorem | TopologicalSpace.vietoris.closure_finite_subsets | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Set",
"closure",
"subset_antisymm",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_iff {f : α → Set β} :
Continuous f ↔ (∀ U, IsOpen U → IsOpen (f ⁻¹' U.powerset)) ∧
(∀ F, IsClosed F → IsClosed (f ⁻¹' F.powerset)) | by
refine ⟨fun hf =>
⟨fun U hU => hU.powerset_vietoris.preimage hf, fun F hF => hF.powerset_vietoris.preimage hf⟩,
fun ⟨h₁, h₂⟩ => ?_⟩
rw [continuous_generateFrom_iff]
rintro _ (⟨U, hU, rfl⟩ | ⟨U, hU, rfl⟩)
· exact h₁ U hU
· simp_rw [preimage_setOf_eq, ← not_disjoint_iff_nonempty_inter, ← compl_setO... | theorem | TopologicalSpace.vietoris.continuous_iff | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Continuous",
"IsClosed",
"IsOpen",
"Set",
"continuous_generateFrom_iff",
"isOpen_compl_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isEmbedding_singleton : IsEmbedding ({·} : α → Set α) | where
injective := Set.singleton_injective
eq_induced := by
simp_rw [TopologicalSpace.vietoris, induced_generateFrom_eq, image_union, image_image,
powerset, preimage_setOf_eq, singleton_subset_iff, singleton_inter_nonempty, union_self,
setOf_mem_eq, image_id', generateFrom_setOf_isOpen] | theorem | TopologicalSpace.vietoris.isEmbedding_singleton | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Set",
"Set.singleton_injective",
"TopologicalSpace.vietoris",
"induced_generateFrom_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_singleton : Continuous ({·} : α → Set α) | isEmbedding_singleton.continuous | theorem | TopologicalSpace.vietoris.continuous_singleton | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Continuous",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.TopologicalSpace.isClosed_range_singleton [T2Space α] {t : TopologicalSpace (Set α)}
(h₁ : IsOpen {(∅ : Set α)}) (h₂ : ∀ {U : Set α}, IsOpen U → IsOpen {s | (s ∩ U).Nonempty}) :
IsClosed (Set.range ({·} : α → Set α)) | by
rw [← isOpen_compl_iff, isOpen_iff_mem_nhds]
intro s hs
rcases Set.eq_empty_or_nonempty s with rfl | h
· rwa [(isOpen_singleton_iff_nhds_eq_pure _).mp h₁, Filter.mem_pure]
rcases h.exists_eq_singleton_or_nontrivial with ⟨x, rfl⟩ | ⟨x, hx, y, hy, hxy⟩
· cases hs <| Set.mem_range_self x
obtain ⟨U, V, hU,... | theorem | TopologicalSpace.isClosed_range_singleton | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Filter.mem_pure",
"IsClosed",
"IsOpen",
"Set",
"Set.eq_empty_or_nonempty",
"Set.mem_range_self",
"Set.mem_setOf",
"Set.range",
"Set.singleton_inter_nonempty",
"T2Space",
"TopologicalSpace",
"isOpen_compl_iff",
"isOpen_iff_mem_nhds",
"isOpen_singleton_iff_nhds_eq_pure",
"t2_separation"
] | Auxiliary lemma showing that singleton sets form a closed set. It takes the required topological
properties as arguments, so that it applies to both the Vietoris topology and the Hausdorff
uniformity. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isClosedEmbedding_singleton [T2Space α] :
Topology.IsClosedEmbedding ({·} : α → Set α) | where
__ := isEmbedding_singleton
isClosed_range :=
isClosed_range_singleton isClopen_singleton_empty.isOpen isOpen_inter_nonempty_of_isOpen | theorem | TopologicalSpace.vietoris.isClosedEmbedding_singleton | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Set",
"T2Space",
"Topology.IsClosedEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_union : Continuous (fun x : Set α × Set α => x.1 ∪ x.2) | by
simp_rw [continuous_iff, powerset, preimage_setOf_eq, union_subset_iff, setOf_and]
exact ⟨
fun U hU => .inter
(hU.powerset_vietoris.preimage continuous_fst)
(hU.powerset_vietoris.preimage continuous_snd),
fun F hF => .inter
(hF.powerset_vietoris.preimage continuous_fst)
(hF.powers... | theorem | TopologicalSpace.vietoris.continuous_union | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Continuous",
"Set",
"continuous_fst",
"continuous_snd"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_range_of_finite {ι : Type*} [Finite ι] :
Continuous (range : (ι → α) → Set α) | by
simp_rw [continuous_iff, powerset, preimage_setOf_eq, range_subset_iff, setOf_forall]
exact ⟨
fun U hU => isOpen_iInter_of_finite fun i => hU.preimage <| continuous_apply i,
fun F hF => isClosed_iInter fun i => hF.preimage <| continuous_apply i⟩ | theorem | TopologicalSpace.vietoris.continuous_range_of_finite | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Continuous",
"Finite",
"Set",
"continuous_apply",
"isClosed_iInter",
"isOpen_iInter_of_finite"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Continuous.image_vietoris (hf : Continuous f) : Continuous (f '' ·) | by
simp_rw [continuous_iff, powerset, preimage_setOf_eq, image_subset_iff]
constructor <;> exact fun U hU => (hU.preimage hf).powerset_vietoris | theorem | Continuous.image_vietoris | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Continuous"
] | When `Set` is equipped with the Vietoris topology, taking the image under a continuous map is
continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_root_.Topology.IsInducing.image_vietoris (hf : IsInducing f) : IsInducing (f '' ·) | by
constructor
have : {U : Set α | IsOpen U} = (f ⁻¹' ·) '' {V : Set β | IsOpen V} :=
Set.ext fun _ => hf.isOpen_iff
simp_rw [TopologicalSpace.vietoris, this, induced_generateFrom_eq, image_union, image_image,
powerset, preimage_setOf_eq, image_subset_iff, image_inter_nonempty_iff] | theorem | Topology.IsInducing.image_vietoris | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsOpen",
"Set",
"Set.ext",
"TopologicalSpace.vietoris",
"induced_generateFrom_eq"
] | When `Set` is equipped with the Vietoris topology, taking the image under an inducing map is
inducing. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_root_.Topology.IsEmbedding.image_vietoris (hf : IsEmbedding f) : IsEmbedding (f '' ·) | where
__ := hf.isInducing.image_vietoris
injective := hf.injective.image_injective | theorem | Topology.IsEmbedding.image_vietoris | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [] | When `Set` is equipped with the Vietoris topology, taking the image under an embedding is an
embedding. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isCompact_aux {K : Set α} (hK : IsCompact K)
{s : Set (Set α)} (hsK : s ⊆ K.powerset) (hs : ∀ L ∈ s, IsCompact L) :
IsCompact {t ⊆ K | ∀ L ∈ s, (t ∩ L).Nonempty} | by
-- By Alexander's subbasis theorem, it is enough to consider covers by the generating sets.
refine isCompact_generateFrom rfl fun S hS hKS => ?_
let u := {U | IsOpen U ∧ {s | (s ∩ U).Nonempty} ∈ S}
by_cases! hsu : ∃ L ∈ s, L ⊆ ⋃₀ u
· /- If the open sets `Uⱼ` in the hit conditions `t ∩ Uⱼ ≠ ∅` cover some `L... | theorem | TopologicalSpace.vietoris.isCompact_aux | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsCompact",
"IsOpen",
"Set",
"isCompact_generateFrom"
] | Given compact sets `K` and `Lᵢ ⊆ K`, the compact subsets of `K` intersecting every `Lᵢ` form a
compact set. This is an auxiliary result used for proving the local compactness of `Compacts α`
without assuming `T2Space α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_root_.IsCompact.powerset_vietoris {K : Set α} (hK : IsCompact K) :
IsCompact K.powerset | by
simpa using! isCompact_aux hK (s := ∅) | theorem | IsCompact.powerset_vietoris | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsCompact",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
subset_closure_of_specializes {s t : Set α} (h : s ⤳ t) : t ⊆ closure s | h.mem_closed isClosed_closure.powerset_vietoris subset_closure | theorem | TopologicalSpace.vietoris.subset_closure_of_specializes | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Set",
"closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
specializes_iff {s t : Set α} : s ⤳ t ↔ (∀ x ∈ s, ∃ y ∈ t, x ⤳ y) ∧ t ⊆ closure s | by
refine ⟨fun h => ⟨fun x hx => ?_, subset_closure_of_specializes h⟩, fun ⟨hst, hts⟩ => ?_⟩
· obtain ⟨y, hyt, hxy⟩ := h.mem_closed (s := {u | (u ∩ closure {x}).Nonempty})
(isClosed_inter_nonempty_of_isClosed isClosed_closure) ⟨x, hx, subset_closure rfl⟩
exact ⟨y, hyt, specializes_iff_mem_closure.mpr hxy⟩... | theorem | TopologicalSpace.vietoris.specializes_iff | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Set",
"Specializes",
"closure",
"isClosed_closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
specializes_iff_of_t1Space {s t : Set α} [T1Space α] : s ⤳ t ↔ s ⊆ t ∧ t ⊆ closure s | by
simp_rw [specializes_iff, specializes_iff_eq, existsAndEq, and_true, ← subset_def] | theorem | TopologicalSpace.vietoris.specializes_iff_of_t1Space | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Set",
"T1Space",
"closure",
"specializes_iff_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
subset_of_specializes {s t : Set α} [T1Space α] (h : s ⤳ t) : s ⊆ t | (specializes_iff_of_t1Space.mp h).1 | theorem | TopologicalSpace.vietoris.subset_of_specializes | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Set",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
specializes_of_subset_closure {s t : Set α} (hst : s ⊆ t) (hts : t ⊆ closure s) :
s ⤳ t | by
aesop (add simp specializes_iff) | theorem | TopologicalSpace.vietoris.specializes_of_subset_closure | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Set",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
specializes_closure {s : Set α} : s ⤳ closure s | specializes_of_subset_closure subset_closure .rfl | theorem | TopologicalSpace.vietoris.specializes_closure | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Set",
"closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
topology : TopologicalSpace (Compacts α) | .induced (↑) (.vietoris α) | instance | TopologicalSpace.Compacts.topology | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"TopologicalSpace"
] | The Vietoris topology on the compact subsets of a topological space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isEmbedding_coe : IsEmbedding ((↑) : Compacts α → Set α) | where
injective := SetLike.coe_injective
eq_induced := rfl | theorem | TopologicalSpace.Compacts.isEmbedding_coe | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Set",
"SetLike.coe_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_coe : Continuous ((↑) : Compacts α → Set α) | isEmbedding_coe.continuous | theorem | TopologicalSpace.Compacts.continuous_coe | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Continuous",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_subsets_of_isOpen {U : Set α} (h : IsOpen U) :
IsOpen {K : Compacts α | ↑K ⊆ U} | continuous_coe.isOpen_preimage _ h.powerset_vietoris | theorem | TopologicalSpace.Compacts.isOpen_subsets_of_isOpen | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_inter_nonempty_of_isOpen {U : Set α} (h : IsOpen U) :
IsOpen {K : Compacts α | (↑K ∩ U).Nonempty} | continuous_coe.isOpen_preimage _ <| vietoris.isOpen_inter_nonempty_of_isOpen h | theorem | TopologicalSpace.Compacts.isOpen_inter_nonempty_of_isOpen | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_subsets_of_isClosed {F : Set α} (h : IsClosed F) :
IsClosed {K : Compacts α | ↑K ⊆ F} | by
simp_rw [← isOpen_compl_iff, Set.compl_setOf, ← Set.inter_compl_nonempty_iff]
exact isOpen_inter_nonempty_of_isOpen h.isOpen_compl | theorem | TopologicalSpace.Compacts.isClosed_subsets_of_isClosed | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsClosed",
"Set",
"Set.compl_setOf",
"Set.inter_compl_nonempty_iff",
"isOpen_compl_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_inter_nonempty_of_isClosed {F : Set α} (h : IsClosed F) :
IsClosed {K : Compacts α | (↑K ∩ F).Nonempty} | by
simp_rw +singlePass [← compl_compl F, Set.inter_compl_nonempty_iff, ← Set.compl_setOf]
exact (isOpen_subsets_of_isOpen h.isOpen_compl).isClosed_compl | theorem | TopologicalSpace.Compacts.isClosed_inter_nonempty_of_isClosed | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsClosed",
"Set",
"Set.compl_setOf",
"Set.inter_compl_nonempty_iff",
"compl_compl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClopen_singleton_bot : IsClopen {(⊥ : Compacts α)} | by
convert! vietoris.isClopen_singleton_empty.preimage continuous_coe
rw [← coe_bot, ← image_singleton (f := SetLike.coe), SetLike.coe_injective.preimage_image] | theorem | TopologicalSpace.Compacts.isClopen_singleton_bot | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsClopen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_finite_subsets (s : Set α) :
closure {K : Compacts α | (K : Set α).Finite ∧ ↑K ⊆ s} = {K : Compacts α | ↑K ⊆ closure s} | by
change closure (SetLike.coe ⁻¹' {K : Set α | K.Finite ∧ K ⊆ s}) =
SetLike.coe ⁻¹' (closure s).powerset
rw [isEmbedding_coe.closure_eq_preimage_closure_image, image_preimage_eq_of_subset ?_,
vietoris.closure_finite_subsets]
exact fun K ⟨hK, _⟩ => ⟨⟨K, hK.isCompact⟩, rfl⟩ | theorem | TopologicalSpace.Compacts.closure_finite_subsets | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Finite",
"Set",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dense_setOf_finite : Dense {K : Compacts α | (K : Set α).Finite} | by
simpa [dense_iff_closure_eq] using closure_finite_subsets univ | theorem | TopologicalSpace.Compacts.dense_setOf_finite | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Dense",
"Finite",
"Set",
"dense_iff_closure_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.TopologicalSpace.IsTopologicalBasis.compacts
{B : Set (Set α)} (hB : IsTopologicalBasis B) :
IsTopologicalBasis <|
(fun u => {K : Compacts α | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
{u | u.Finite ∧ u ⊆ B} | by
refine hB.vietoris.isInducing isEmbedding_coe.isInducing
|>.isTopologicalBasis_of_exists_subset ?_ ?_ <;> simp_rw [forall_mem_image]
· intro u ⟨hu, huB⟩
simp_rw [setOf_and, setOf_forall]
exact .inter
(isOpen_subsets_of_isOpen <| isOpen_sUnion fun U hU => hB.isOpen <| huB hU)
(hu.isOpen_bi... | theorem | TopologicalSpace.IsTopologicalBasis.compacts | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Set",
"isOpen_sUnion"
] | Given a basis `B` on a topological space `α`, the topology of `Compacts α` has a basis
consisting of sets of the form `{K | K ⊆ U₁ ∪ … ∪ Uₙ, K ∩ U₁ ≠ ∅, …, K ∩ Uₙ ≠ ∅}`, where
`U₁, …, Uₙ ∈ B`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isTopologicalBasis :
IsTopologicalBasis <|
(fun u => {K : Compacts α | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
{u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen U} | isTopologicalBasis_opens.compacts | theorem | TopologicalSpace.Compacts.isTopologicalBasis | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsOpen",
"Set"
] | The topology of `Compacts α` has a basis consisting of sets of the form
`{K | K ⊆ U₁ ∪ … ∪ Uₙ, K ∩ U₁ ≠ ∅, …, K ∩ Uₙ ≠ ∅}`, where `U₁, …, Uₙ` are open sets.
See also `TopologicalSpace.IsTopologicalBasis.compacts` for a variant where the sets `Uᵢ`
are chosen from some basis. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isEmbedding_singleton : IsEmbedding ({·} : α → Compacts α) | isEmbedding_coe.of_comp_iff.mp vietoris.isEmbedding_singleton | theorem | TopologicalSpace.Compacts.isEmbedding_singleton | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_singleton : Continuous ({·} : α → Compacts α) | isEmbedding_singleton.continuous | theorem | TopologicalSpace.Compacts.continuous_singleton | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosedEmbedding_singleton [T2Space α] :
Topology.IsClosedEmbedding ({·} : α → Compacts α) | .of_comp isEmbedding_coe vietoris.isClosedEmbedding_singleton | theorem | TopologicalSpace.Compacts.isClosedEmbedding_singleton | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"T2Space",
"Topology.IsClosedEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_prod : Continuous fun p : Compacts α × Compacts β => p.1 ×ˢ p.2 | by
rw [continuous_induced_rng, continuous_generateFrom_iff]
rintro _ (⟨U, hU, rfl⟩ | ⟨U, hU, rfl⟩)
· rw [isOpen_iff_forall_mem_open]
intro ⟨K, L⟩ (hKL : (K : Set α) ×ˢ (L : Set β) ⊆ U)
obtain ⟨V, W, hV, hW, hK, hL, hVW⟩ := generalized_tube_lemma K.isCompact L.isCompact hU hKL
exact ⟨_, fun ⟨K', L'⟩ ⟨h... | theorem | TopologicalSpace.Compacts.continuous_prod | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Continuous",
"Set",
"Set.prod_mono",
"continuous_generateFrom_iff",
"continuous_induced_rng",
"generalized_tube_lemma",
"isOpen_iff_forall_mem_open",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Continuous.compacts_map (hf : Continuous f) : Continuous (Compacts.map f hf) | isEmbedding_coe.continuous_iff.mpr <| hf.image_vietoris.comp continuous_coe | theorem | Continuous.compacts_map | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Topology.IsInducing.compacts_map (hf : IsInducing f) :
IsInducing (Compacts.map f hf.continuous) | isEmbedding_coe.isInducing.of_comp_iff.mp <| hf.image_vietoris.comp isEmbedding_coe.isInducing | theorem | Topology.IsInducing.compacts_map | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Topology.IsEmbedding.compacts_map (hf : IsEmbedding f) :
IsEmbedding (Compacts.map f hf.continuous) | isEmbedding_coe.of_comp_iff.mp <| hf.image_vietoris.comp isEmbedding_coe | theorem | Topology.IsEmbedding.compacts_map | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Topology.IsOpenEmbedding.compacts_map (hf : IsOpenEmbedding f) :
IsOpenEmbedding (Compacts.map f hf.continuous) | where
__ := hf.isEmbedding.compacts_map
isOpen_range := range_map hf.isInducing ▸ isOpen_subsets_of_isOpen hf.isOpen_range | theorem | Topology.IsOpenEmbedding.compacts_map | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Topology.IsClosedEmbedding.compacts_map (hf : IsClosedEmbedding f) :
IsClosedEmbedding (Compacts.map f hf.continuous) | where
__ := hf.isEmbedding.compacts_map
isClosed_range := range_map hf.isInducing ▸ isClosed_subsets_of_isClosed hf.isClosed_range | theorem | Topology.IsClosedEmbedding.compacts_map | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
discreteTopology_iff : DiscreteTopology (Compacts α) ↔ DiscreteTopology α | ⟨fun _ => isEmbedding_singleton.discreteTopology, fun _ => inferInstance⟩ | theorem | TopologicalSpace.Compacts.discreteTopology_iff | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"DiscreteTopology"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t2Space_iff : T2Space (Compacts α) ↔ T2Space α | ⟨fun _ => isEmbedding_singleton.t2Space, fun _ => inferInstance⟩ | theorem | TopologicalSpace.Compacts.t2Space_iff | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
regularSpace_iff : RegularSpace (Compacts α) ↔ RegularSpace α | ⟨fun _ => isEmbedding_singleton.regularSpace, fun _ => inferInstance⟩ | theorem | TopologicalSpace.Compacts.regularSpace_iff | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"RegularSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t3Space_iff : T3Space (Compacts α) ↔ T3Space α | ⟨fun _ => isEmbedding_singleton.t3Space, fun _ => inferInstance⟩ | theorem | TopologicalSpace.Compacts.t3Space_iff | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"T3Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isCompact_subsets_of_isCompact {K : Set α} (hK : IsCompact K) :
IsCompact {L : Compacts α | ↑L ⊆ K} | by
rw [isEmbedding_coe.isCompact_iff]
refine .of_subset_of_specializes hK.powerset_vietoris (by grind) (fun s hs => ?_)
let L : Compacts α := ⟨K ∩ closure s, hK.inter_right isClosed_closure⟩
exact ⟨L, mem_image_of_mem _ inter_subset_left,
vietoris.specializes_of_subset_closure (subset_inter hs subset_closur... | theorem | TopologicalSpace.Compacts.isCompact_subsets_of_isCompact | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsCompact",
"Set",
"closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isCompact_biUnion_coe_of_isCompact {S : Set (Compacts α)} (hS : IsCompact S) :
IsCompact (⋃ K ∈ S, (K : Set α)) | by
rw [isCompact_iff_finite_subcover]
intro ι U hU h
rw [iUnion₂_subset_iff] at h
obtain ⟨s, hs⟩ := hS.elim_finite_subcover
(fun s : Finset ι => {K | (K : Set α) ⊆ ⋃ i ∈ s, U i})
(fun s => isOpen_subsets_of_isOpen <| isOpen_biUnion fun _ _ => hU _)
(fun K hK => mem_iUnion.mpr <| K.isCompact.elim_fin... | theorem | TopologicalSpace.Compacts.isCompact_biUnion_coe_of_isCompact | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Finset",
"Finset.set_biUnion_biUnion",
"IsCompact",
"Set",
"isCompact_iff_finite_subcover",
"isOpen_biUnion",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compactSpace_iff : CompactSpace (Compacts α) ↔ CompactSpace α | by
refine ⟨fun h => ⟨?_⟩, fun _ => inferInstance⟩
convert! isCompact_biUnion_coe_of_isCompact (α := α) isCompact_univ
symm
simp_rw [biUnion_univ, eq_univ_iff_forall, mem_iUnion]
exact fun x => ⟨{x}, Set.mem_singleton x⟩ | theorem | TopologicalSpace.Compacts.compactSpace_iff | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"CompactSpace",
"Set.mem_singleton",
"isCompact_univ",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
noncompactSpace_iff : NoncompactSpace (Compacts α) ↔ NoncompactSpace α | by
simp_rw [← not_compactSpace_iff, compactSpace_iff] | theorem | TopologicalSpace.Compacts.noncompactSpace_iff | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"NoncompactSpace",
"not_compactSpace_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
topology : TopologicalSpace (NonemptyCompacts α) | .induced (↑) (.vietoris α) | instance | TopologicalSpace.NonemptyCompacts.topology | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"TopologicalSpace"
] | The Vietoris topology on the nonempty compact subsets of a topological space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isEmbedding_coe : IsEmbedding ((↑) : NonemptyCompacts α → Set α) | where
injective := SetLike.coe_injective
eq_induced := rfl | theorem | TopologicalSpace.NonemptyCompacts.isEmbedding_coe | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Set",
"SetLike.coe_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_coe : Continuous ((↑) : NonemptyCompacts α → Set α) | isEmbedding_coe.continuous | theorem | TopologicalSpace.NonemptyCompacts.continuous_coe | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Continuous",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isEmbedding_toCompacts : IsEmbedding (toCompacts (α := α)) | where
injective := toCompacts_injective
eq_induced := .symm <| induced_compose (f := toCompacts) (g := SetLike.coe) | theorem | TopologicalSpace.NonemptyCompacts.isEmbedding_toCompacts | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"induced_compose",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_toCompacts : Continuous (toCompacts (α := α)) | isEmbedding_toCompacts.continuous | theorem | TopologicalSpace.NonemptyCompacts.continuous_toCompacts | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosedEmbedding_toCompacts : IsClosedEmbedding (toCompacts (α := α)) | where
__ := isEmbedding_toCompacts
isClosed_range := by
rw [range_toCompacts]
exact Compacts.isClopen_singleton_bot.compl.isClosed | theorem | TopologicalSpace.NonemptyCompacts.isClosedEmbedding_toCompacts | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpenEmbedding_toCompacts : IsOpenEmbedding (toCompacts (α := α)) | where
__ := isEmbedding_toCompacts
isOpen_range := by
rw [range_toCompacts]
exact Compacts.isClopen_singleton_bot.compl.isOpen | theorem | TopologicalSpace.NonemptyCompacts.isOpenEmbedding_toCompacts | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_subsets_of_isOpen {U : Set α} (h : IsOpen U) :
IsOpen {K : NonemptyCompacts α | ↑K ⊆ U} | h.powerset_vietoris.preimage continuous_coe | theorem | TopologicalSpace.NonemptyCompacts.isOpen_subsets_of_isOpen | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_inter_nonempty_of_isOpen {U : Set α} (h : IsOpen U) :
IsOpen {K : NonemptyCompacts α | (↑K ∩ U).Nonempty} | (vietoris.isOpen_inter_nonempty_of_isOpen h).preimage continuous_coe | theorem | TopologicalSpace.NonemptyCompacts.isOpen_inter_nonempty_of_isOpen | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_subsets_of_isClosed {F : Set α} (h : IsClosed F) :
IsClosed {K : NonemptyCompacts α | ↑K ⊆ F} | h.powerset_vietoris.preimage continuous_coe | theorem | TopologicalSpace.NonemptyCompacts.isClosed_subsets_of_isClosed | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsClosed",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_inter_nonempty_of_isClosed {F : Set α} (h : IsClosed F) :
IsClosed {K : NonemptyCompacts α | (↑K ∩ F).Nonempty} | (vietoris.isClosed_inter_nonempty_of_isClosed h).preimage continuous_coe | theorem | TopologicalSpace.NonemptyCompacts.isClosed_inter_nonempty_of_isClosed | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsClosed",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_finite_subsets (s : Set α) :
closure {K : NonemptyCompacts α | (K : Set α).Finite ∧ ↑K ⊆ s} =
{K : NonemptyCompacts α | ↑K ⊆ closure s} | by
simpa [isOpenEmbedding_toCompacts.isOpenMap.preimage_closure_eq_closure_preimage
continuous_toCompacts] using congr(toCompacts ⁻¹' $(Compacts.closure_finite_subsets s)) | theorem | TopologicalSpace.NonemptyCompacts.closure_finite_subsets | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Finite",
"Set",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dense_setOf_finite : Dense {K : NonemptyCompacts α | (K : Set α).Finite} | Compacts.dense_setOf_finite.preimage isOpenEmbedding_toCompacts.isOpenMap | theorem | TopologicalSpace.NonemptyCompacts.dense_setOf_finite | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Dense",
"Finite",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.TopologicalSpace.IsTopologicalBasis.nonemptyCompacts
{B : Set (Set α)} (hB : IsTopologicalBasis B) :
IsTopologicalBasis <|
(fun u => {K : NonemptyCompacts α | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
{u | u.Finite ∧ u.Nonempty ∧ u ⊆ B} | by
refine hB.compacts.isInducing isEmbedding_toCompacts.isInducing
|>.isTopologicalBasis_of_exists_subset ?_ ?_ <;> simp_rw [forall_mem_image]
· rintro u ⟨hu, -, huB⟩
simp_rw [setOf_and, setOf_forall]
exact .inter
(isOpen_subsets_of_isOpen <| isOpen_sUnion fun U hU => hB.isOpen <| huB hU)
(h... | theorem | TopologicalSpace.IsTopologicalBasis.nonemptyCompacts | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"Set",
"isOpen_sUnion"
] | Given a basis `B` on a topological space `α`, the topology of `NonemptyCompacts α` has a basis
consisting of sets of the form `{K | K ⊆ U₁ ∪ … ∪ Uₙ, K ∩ U₁ ≠ ∅, …, K ∩ Uₙ ≠ ∅}`, where
`U₁, …, Uₙ ∈ B` and `n > 0`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isTopologicalBasis :
IsTopologicalBasis <|
(fun u => {K : NonemptyCompacts α | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
{u : Set (Set α) | u.Finite ∧ u.Nonempty ∧ ∀ U ∈ u, IsOpen U} | isTopologicalBasis_opens.nonemptyCompacts | theorem | TopologicalSpace.NonemptyCompacts.isTopologicalBasis | Topology.Sets | Mathlib/Topology/Sets/VietorisTopology.lean | [] | [
"IsOpen",
"Set"
] | The topology of `NonemptyCompacts α` has a basis consisting of sets of the form
`{s | s ⊆ U₁ ∪ … ∪ Uₙ, s ∩ U₁ ≠ ∅, …, s ∩ Uₙ ≠ ∅}`, where `U₁, …, Uₙ` are open sets and `n > 0`.
See also `TopologicalSpace.IsTopologicalBasis.nonemptyCompacts` for a variant where the sets `Uᵢ`
are chosen from some basis. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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