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Closeds (α : Type*) [TopologicalSpace α] where
/-- the carrier set, i.e. the points in this set -/
carrier : Set α
isClosed' : IsClosed carrier | structure | TopologicalSpace.Closeds | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsClosed",
"Set",
"TopologicalSpace"
] | The type of closed subsets of a topological space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed (s : Closeds α) : IsClosed (s : Set α) | s.isClosed' | theorem | TopologicalSpace.Closeds.isClosed | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsClosed",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Simps.coe (s : Closeds α) : Set α | s
initialize_simps_projections Closeds (carrier → coe, as_prefix coe) | def | TopologicalSpace.Closeds.Simps.coe | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | See Note [custom simps projection]. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
carrier_eq_coe (s : Closeds α) : s.carrier = (s : Set α) | rfl | lemma | TopologicalSpace.Closeds.carrier_eq_coe | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ext {s t : Closeds α} (h : (s : Set α) = t) : s = t | SetLike.ext' h | theorem | TopologicalSpace.Closeds.ext | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set",
"SetLike.ext'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_mk (s : Set α) (h) : (mk s h : Set α) = s | rfl | theorem | TopologicalSpace.Closeds.coe_mk | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_mk {s : Set α} {hs : IsClosed s} {x : α} : x ∈ (⟨s, hs⟩ : Closeds α) ↔ x ∈ s | .rfl | lemma | TopologicalSpace.Closeds.mem_mk | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsClosed",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure (s : Set α) : Closeds α | ⟨closure s, isClosed_closure⟩ | def | TopologicalSpace.Closeds.closure | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set",
"closure"
] | The closure of a set, as an element of `TopologicalSpace.Closeds`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_closure {s : Set α} {x : α} : x ∈ Closeds.closure s ↔ x ∈ closure s | .rfl | theorem | TopologicalSpace.Closeds.mem_closure | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
gc : GaloisConnection Closeds.closure ((↑) : Closeds α → Set α) | fun _ U =>
⟨subset_closure.trans, fun h => closure_minimal h U.isClosed⟩ | theorem | TopologicalSpace.Closeds.gc | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"GaloisConnection",
"Set",
"closure_minimal"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_le {s : Set α} {t : Closeds α} : .closure s ≤ t ↔ s ⊆ t | t.isClosed.closure_subset_iff | lemma | TopologicalSpace.Closeds.closure_le | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.Closeds.gi | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"GaloisInsertion",
"SetLike.coe_injective",
"subset_closure"
] | The Galois insertion between sets and closeds. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
instCompleteLattice : CompleteLattice (Closeds α) | fast_instance% CompleteLattice.copy
(GaloisInsertion.liftCompleteLattice gi)
-- le
_ rfl
-- top
⟨univ, isClosed_univ⟩ rfl
-- bot
⟨∅, isClosed_empty⟩ (SetLike.coe_injective closure_empty.symm)
-- sup
(fun s t => ⟨s ∪ t, s.2.union t.2⟩)
(funext fun s => funext fun t => SetLike.coe_... | instance | TopologicalSpace.Closeds.instCompleteLattice | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"CompleteLattice",
"CompleteLattice.copy",
"GaloisInsertion.liftCompleteLattice",
"SetLike.coe_injective",
"isClosed_biInter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_sup (s t : Closeds α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t | by
rfl | theorem | TopologicalSpace.Closeds.coe_sup | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_inf (s t : Closeds α) : (↑(s ⊓ t) : Set α) = ↑s ∩ ↑t | rfl | theorem | TopologicalSpace.Closeds.coe_inf | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_top : (↑(⊤ : Closeds α) : Set α) = univ | rfl | theorem | TopologicalSpace.Closeds.coe_top | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_eq_univ {s : Closeds α} : (s : Set α) = univ ↔ s = ⊤ | SetLike.coe_injective.eq_iff' rfl | theorem | TopologicalSpace.Closeds.coe_eq_univ | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_bot : (↑(⊥ : Closeds α) : Set α) = ∅ | rfl | theorem | TopologicalSpace.Closeds.coe_bot | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_eq_empty {s : Closeds α} : (s : Set α) = ∅ ↔ s = ⊥ | SetLike.coe_injective.eq_iff' rfl | theorem | TopologicalSpace.Closeds.coe_eq_empty | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_nonempty {s : Closeds α} : (s : Set α).Nonempty ↔ s ≠ ⊥ | nonempty_iff_ne_empty.trans coe_eq_empty.not | theorem | TopologicalSpace.Closeds.coe_nonempty | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_sInf {S : Set (Closeds α)} : (↑(sInf S) : Set α) = ⋂ i ∈ S, ↑i | rfl | theorem | TopologicalSpace.Closeds.coe_sInf | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_sSup {S : Set (Closeds α)} : ((sSup S : Closeds α) : Set α) =
closure (⋃₀ ((↑) '' S)) | by rfl | lemma | TopologicalSpace.Closeds.coe_sSup | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_finset_sup (f : ι → Closeds α) (s : Finset ι) :
(↑(s.sup f) : Set α) = s.sup ((↑) ∘ f) | map_finset_sup (⟨⟨(↑), coe_sup⟩, coe_bot⟩ : SupBotHom (Closeds α) (Set α)) _ _ | theorem | TopologicalSpace.Closeds.coe_finset_sup | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Finset",
"Set",
"SupBotHom",
"map_finset_sup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_finset_inf (f : ι → Closeds α) (s : Finset ι) :
(↑(s.inf f) : Set α) = s.inf ((↑) ∘ f) | map_finset_inf (⟨⟨(↑), coe_inf⟩, coe_top⟩ : InfTopHom (Closeds α) (Set α)) _ _ | theorem | TopologicalSpace.Closeds.coe_finset_inf | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Finset",
"InfTopHom",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_sInf {S : Set (Closeds α)} {x : α} : x ∈ sInf S ↔ ∀ s ∈ S, x ∈ s | mem_iInter₂ | theorem | TopologicalSpace.Closeds.mem_sInf | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_iInf {ι} {x : α} {s : ι → Closeds α} : x ∈ iInf s ↔ ∀ i, x ∈ s i | by simp [iInf] | theorem | TopologicalSpace.Closeds.mem_iInf | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_iInf {ι} (s : ι → Closeds α) : ((⨅ i, s i : Closeds α) : Set α) = ⋂ i, s i | by
ext; simp | theorem | TopologicalSpace.Closeds.coe_iInf | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iInf_def {ι} (s : ι → Closeds α) :
⨅ i, s i = ⟨⋂ i, s i, isClosed_iInter fun i => (s i).2⟩ | by ext1; simp | theorem | TopologicalSpace.Closeds.iInf_def | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"isClosed_iInter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iInf_mk {ι} (s : ι → Set α) (h : ∀ i, IsClosed (s i)) :
(⨅ i, ⟨s i, h i⟩ : Closeds α) = ⟨⋂ i, s i, isClosed_iInter h⟩ | iInf_def _ | theorem | TopologicalSpace.Closeds.iInf_mk | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsClosed",
"Set",
"isClosed_iInter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.Closeds.coframeMinimalAxioms | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set.union_iInter₂",
"SetLike.coe_injective"
] | Closed sets in a topological space form a coframe. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
instCoframe : Coframe (Closeds α) | fast_instance% .ofMinimalAxioms coframeMinimalAxioms | instance | TopologicalSpace.Closeds.instCoframe | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"ofMinimalAxioms"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
singleton [T1Space α] (x : α) : Closeds α | {x} | abbrev | TopologicalSpace.Closeds.singleton | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"T1Space"
] | The term of `TopologicalSpace.Closeds α` corresponding to a singleton. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mk_singleton [T1Space α] {x : α} :
(⟨{x}, isClosed_singleton⟩ : Closeds α) = {x} | rfl | theorem | TopologicalSpace.Closeds.mk_singleton | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_singleton [T1Space α] {a b : α} : a ∈ ({b} : Closeds α) ↔ a = b | Iff.rfl | lemma | TopologicalSpace.Closeds.mem_singleton | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
singleton_injective [T1Space α] : Function.Injective ({·} : α → Closeds α) | .of_comp (f := SetLike.coe) Set.singleton_injective | theorem | TopologicalSpace.Closeds.singleton_injective | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set.singleton_injective",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
singleton_inj [T1Space α] {x y : α} : ({x} : Closeds α) = {y} ↔ x = y | singleton_injective.eq_iff | theorem | TopologicalSpace.Closeds.singleton_inj | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
preimage (s : Closeds β) {f : α → β} (hf : Continuous f) : Closeds α | ⟨f ⁻¹' s, s.isClosed.preimage hf⟩ | def | TopologicalSpace.Closeds.preimage | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Continuous"
] | The preimage of a closed set under a continuous map. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_prod (s : Closeds α) (t : Closeds β) :
(s ×ˢ t : Closeds (α × β)) = (s : Set α) ×ˢ (t : Set β) | rfl | theorem | TopologicalSpace.Closeds.coe_prod | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_prod {s : Closeds α} {t : Closeds β} {x : α × β} : x ∈ s ×ˢ t ↔ x.1 ∈ s ∧ x.2 ∈ t | Iff.rfl | theorem | TopologicalSpace.Closeds.mem_prod | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
singleton_prod_singleton [T1Space α] [T1Space β] (x : α) (y : β) :
({x} ×ˢ {y} : Closeds (α × β)) = {(x, y)} | Closeds.ext Set.singleton_prod_singleton | theorem | TopologicalSpace.Closeds.singleton_prod_singleton | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set.singleton_prod_singleton",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Closeds.compl (s : Closeds α) : Opens α | ⟨sᶜ, s.2.isOpen_compl⟩ | def | TopologicalSpace.Closeds.compl | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [] | The complement of a closed set as an open set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Opens.compl (s : Opens α) : Closeds α | ⟨sᶜ, s.2.isClosed_compl⟩ | def | TopologicalSpace.Opens.compl | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [] | The complement of an open set as a closed set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Closeds.compl_compl (s : Closeds α) : s.compl.compl = s | Closeds.ext (compl_compl (s : Set α)) | theorem | TopologicalSpace.Closeds.compl_compl | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set",
"compl_compl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Opens.compl_compl (s : Opens α) : s.compl.compl = s | Opens.ext (compl_compl (s : Set α)) | theorem | TopologicalSpace.Opens.compl_compl | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set",
"compl_compl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Closeds.compl_bijective : Function.Bijective (@Closeds.compl α _) | Function.bijective_iff_has_inverse.mpr ⟨Opens.compl, Closeds.compl_compl, Opens.compl_compl⟩ | theorem | TopologicalSpace.Closeds.compl_bijective | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Function.Bijective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Opens.compl_bijective : Function.Bijective (@Opens.compl α _) | Function.bijective_iff_has_inverse.mpr ⟨Closeds.compl, Opens.compl_compl, Closeds.compl_compl⟩ | theorem | TopologicalSpace.Opens.compl_bijective | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Function.Bijective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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_compl | def | TopologicalSpace.Closeds.complOrderIso | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"OrderDual.ofDual",
"OrderDual.toDual",
"OrderDual.toDual_le_toDual",
"trans"
] | `TopologicalSpace.Closeds.compl` as an `OrderIso` to the order dual of
`TopologicalSpace.Opens α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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_compl | def | TopologicalSpace.Opens.complOrderIso | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"OrderDual.ofDual",
"OrderDual.toDual",
"OrderDual.toDual_le_toDual",
"trans"
] | `TopologicalSpace.Opens.compl` as an `OrderIso` to the order dual of
`TopologicalSpace.Closeds α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 hts | lemma | TopologicalSpace.Closeds.coe_eq_singleton_of_isAtom | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsAtom",
"Set",
"T0Space",
"minimal_nonempty_closed_eq_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Closeds.isAtom_coe [T1Space α] {s : Closeds α} :
IsAtom (s : Set α) ↔ IsAtom s | Closeds.gi.isAtom_iff' rfl
(fun t ht ↦ by obtain ⟨x, rfl⟩ := Set.isAtom_iff.1 ht; exact closure_singleton) s | lemma | TopologicalSpace.Closeds.isAtom_coe | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsAtom",
"Set",
"T1Space",
"closure_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Closeds.isAtom_iff [T1Space α] {s : Closeds α} :
IsAtom s ↔ ∃ x, s = {x} | by
simp [← Closeds.isAtom_coe, Set.isAtom_iff, SetLike.ext_iff, Set.ext_iff] | theorem | TopologicalSpace.Closeds.isAtom_iff | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsAtom",
"Set.isAtom_iff",
"SetLike.ext_iff",
"T1Space"
] | in a `T1Space`, atoms of `TopologicalSpace.Closeds α` are precisely the singletons. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Opens.isCoatom_iff [T1Space α] {s : Opens α} :
IsCoatom s ↔ ∃ x, s = ({x} : Closeds α).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.eq_iff] | theorem | TopologicalSpace.Opens.isCoatom_iff | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsAtom",
"IsCoatom",
"T1Space",
"isAtom_dual_iff_isCoatom"
] | in a `T1Space`, coatoms of `TopologicalSpace.Opens α` are precisely complements of singletons:
`({x} : Closeds α).compl`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Clopens (α : Type*) [TopologicalSpace α] where
/-- the carrier set, i.e. the points in this set -/
carrier : Set α
isClopen' : IsClopen carrier | structure | TopologicalSpace.Clopens | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsClopen",
"Set",
"TopologicalSpace"
] | The type of clopen sets of a topological space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClopen (s : Clopens α) : IsClopen (s : Set α) | s.isClopen' | theorem | TopologicalSpace.Clopens.isClopen | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsClopen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen (s : Clopens α) : IsOpen (s : Set α) | s.isClopen.isOpen | lemma | TopologicalSpace.Clopens.isOpen | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed (s : Clopens α) : IsClosed (s : Set α) | s.isClopen.isClosed | lemma | TopologicalSpace.Clopens.isClosed | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsClosed",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Simps.coe (s : Clopens α) : Set α | s
initialize_simps_projections Clopens (carrier → coe, as_prefix coe) | def | TopologicalSpace.Clopens.Simps.coe | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | See Note [custom simps projection]. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toOpens (s : Clopens α) : Opens α | ⟨s, s.isOpen⟩ | def | TopologicalSpace.Clopens.toOpens | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [] | Reinterpret a clopen as an open. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toCloseds (s : Clopens α) : Closeds α | ⟨s, s.isClosed⟩ | def | TopologicalSpace.Clopens.toCloseds | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [] | Reinterpret a clopen as a closed. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ext {s t : Clopens α} (h : (s : Set α) = t) : s = t | SetLike.ext' h | theorem | TopologicalSpace.Clopens.ext | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set",
"SetLike.ext'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_mk {s : Set α} {x h} : x ∈ mk s h ↔ x ∈ s | .rfl | lemma | TopologicalSpace.Clopens.mem_mk | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_sup (s t : Clopens α) : ↑(s ⊔ t) = (s ∪ t : Set α) | rfl | lemma | TopologicalSpace.Clopens.coe_sup | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_inf (s t : Clopens α) : ↑(s ⊓ t) = (s ∩ t : Set α) | rfl | lemma | TopologicalSpace.Clopens.coe_inf | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_top : (↑(⊤ : Clopens α) : Set α) = univ | rfl | lemma | TopologicalSpace.Clopens.coe_top | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_bot : (↑(⊥ : Clopens α) : Set α) = ∅ | rfl | lemma | TopologicalSpace.Clopens.coe_bot | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_sdiff (s t : Clopens α) : ↑(s \ t) = (s \ t : Set α) | rfl | lemma | TopologicalSpace.Clopens.coe_sdiff | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_himp (s t : Clopens α) : ↑(s ⇨ t) = (s ⇨ t : Set α) | rfl | lemma | TopologicalSpace.Clopens.coe_himp | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_compl (s : Clopens α) : (↑sᶜ : Set α) = (↑s)ᶜ | rfl | lemma | TopologicalSpace.Clopens.coe_compl | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_prod {s : Clopens α} {t : Clopens β} {x : α × β} :
x ∈ s ×ˢ t ↔ x.1 ∈ s ∧ x.2 ∈ t | .rfl | lemma | TopologicalSpace.Clopens.mem_prod | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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] | lemma | TopologicalSpace.Clopens.coe_finset_sup | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Finset",
"Finset.induction_on",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_disjoint {s t : Clopens α} : Disjoint (s : Set α) t ↔ Disjoint s t | by
simp [disjoint_iff, ← SetLike.coe_set_eq] | lemma | TopologicalSpace.Clopens.coe_disjoint | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Disjoint",
"Set",
"SetLike.coe_set_eq",
"disjoint_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IrreducibleCloseds (α : Type*) [TopologicalSpace α] where
/-- the carrier set, i.e. the points in this set -/
carrier : Set α
isIrreducible' : IsIrreducible carrier
isClosed' : IsClosed carrier | structure | TopologicalSpace.IrreducibleCloseds | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsClosed",
"IsIrreducible",
"Set",
"TopologicalSpace"
] | The type of irreducible closed subsets of a topological space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isIrreducible (s : IrreducibleCloseds α) : IsIrreducible (s : Set α) | s.isIrreducible' | theorem | TopologicalSpace.IrreducibleCloseds.isIrreducible | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsIrreducible",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed (s : IrreducibleCloseds α) : IsClosed (s : Set α) | s.isClosed' | theorem | TopologicalSpace.IrreducibleCloseds.isClosed | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsClosed",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Simps.coe (s : IrreducibleCloseds α) : Set α | s
initialize_simps_projections IrreducibleCloseds (carrier → coe, as_prefix coe) | def | TopologicalSpace.IrreducibleCloseds.Simps.coe | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set"
] | See Note [custom simps projection]. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ext {s t : IrreducibleCloseds α} (h : (s : Set α) = t) : s = t | SetLike.ext' h | theorem | TopologicalSpace.IrreducibleCloseds.ext | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set",
"SetLike.ext'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_mk (s : Set α) (h : IsIrreducible s) (h' : IsClosed s) : (mk s h h' : Set α) = s | rfl | theorem | TopologicalSpace.IrreducibleCloseds.coe_mk | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsClosed",
"IsIrreducible",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
singleton [T1Space α] (x : α) : IrreducibleCloseds α | {x} | abbrev | TopologicalSpace.IrreducibleCloseds.singleton | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"T1Space"
] | The term of `TopologicalSpace.IrreducibleCloseds α` corresponding to a singleton. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mk_singleton [T1Space α] {x : α} :
(⟨{x}, isIrreducible_singleton, isClosed_singleton⟩ : IrreducibleCloseds α) = {x} | rfl | theorem | TopologicalSpace.IrreducibleCloseds.mk_singleton | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"T1Space",
"isIrreducible_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_singleton [T1Space α] {a b : α} : a ∈ ({b} : IrreducibleCloseds α) ↔ a = b | Iff.rfl | lemma | TopologicalSpace.IrreducibleCloseds.mem_singleton | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
singleton_injective [T1Space α] : Function.Injective ({·} : α → IrreducibleCloseds α) | .of_comp (f := SetLike.coe) Set.singleton_injective | theorem | TopologicalSpace.IrreducibleCloseds.singleton_injective | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set.singleton_injective",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
singleton_inj [T1Space α] {x y : α} : ({x} : IrreducibleCloseds α) = {y} ↔ x = y | singleton_injective.eq_iff | theorem | TopologicalSpace.IrreducibleCloseds.singleton_inj | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | TopologicalSpace.IrreducibleCloseds.equivSubtype | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsClosed",
"IsIrreducible",
"Set"
] | The equivalence between `IrreducibleCloseds α` and `{x : Set α // IsIrreducible x ∧ IsClosed x }`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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⟩ | def | TopologicalSpace.IrreducibleCloseds.equivSubtype' | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsClosed",
"IsIrreducible",
"Set"
] | The equivalence between `IrreducibleCloseds α` and `{x : Set α // IsClosed x ∧ IsIrreducible x }`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
orderIsoSubtype : IrreducibleCloseds α ≃o { x : Set α // IsIrreducible x ∧ IsClosed x } | equivSubtype.toOrderIso (fun _ _ h ↦ h) (fun _ _ h ↦ h) | def | TopologicalSpace.IrreducibleCloseds.orderIsoSubtype | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsClosed",
"IsIrreducible",
"Set"
] | The equivalence `IrreducibleCloseds α ≃ { x : Set α // IsIrreducible x ∧ IsClosed x }` is an
order isomorphism. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
orderIsoSubtype' : IrreducibleCloseds α ≃o { x : Set α // IsClosed x ∧ IsIrreducible x } | equivSubtype'.toOrderIso (fun _ _ h ↦ h) (fun _ _ h ↦ h) | def | TopologicalSpace.IrreducibleCloseds.orderIsoSubtype' | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"IsClosed",
"IsIrreducible",
"Set"
] | The equivalence `IrreducibleCloseds α ≃ { x : Set α // IsClosed x ∧ IsIrreducible x }` is an
order isomorphism. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
map (f : β → α) (hf : Continuous f)
(c : IrreducibleCloseds β) : IrreducibleCloseds α | where
carrier := closure (f '' c)
isIrreducible' := c.isIrreducible.image f hf.continuousOn |>.closure
isClosed' := isClosed_closure | def | TopologicalSpace.IrreducibleCloseds.map | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Continuous",
"closure",
"isClosed_closure"
] | The map on irreducible closed sets induced by a continuous map `f`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_map (f : β → α) (hf : Continuous f) (s : IrreducibleCloseds β) :
(map f hf s : Set α) = closure (f '' s) | rfl | lemma | TopologicalSpace.IrreducibleCloseds.coe_map | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Continuous",
"Set",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_mono {f : β → α} (hf : Continuous f) : Monotone (map f hf) | fun _ _ h_le => closure_mono <| Set.image_mono h_le | lemma | TopologicalSpace.IrreducibleCloseds.map_mono | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Continuous",
"Monotone",
"Set.image_mono",
"closure_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_injective_of_isInducing {f : β → α} (hf : IsInducing f) :
Function.Injective (map f hf.continuous) | by
intro A B h_images_eq
apply SetLike.coe_injective
replace h_images_eq : closure (f '' A) = closure (f '' B) := congr($h_images_eq)
rw [← A.isClosed.closure_eq, hf.closure_eq_preimage_closure_image, h_images_eq,
← hf.closure_eq_preimage_closure_image, B.isClosed.closure_eq] | lemma | TopologicalSpace.IrreducibleCloseds.map_injective_of_isInducing | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"SetLike.coe_injective",
"closure"
] | The map `IrreducibleCloseds.map` is injective when `f` is inducing.
This relies on the property of embeddings that a closed set in the domain is the preimage
of the closure of its image. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
map_strictMono_of_isInducing {f : β → α} (hf : IsInducing f) :
StrictMono (map f hf.continuous) | Monotone.strictMono_of_injective (map_mono hf.continuous) (map_injective_of_isInducing hf) | lemma | TopologicalSpace.IrreducibleCloseds.map_strictMono_of_isInducing | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Monotone.strictMono_of_injective",
"StrictMono"
] | The map `IrreducibleCloseds.map` is strictly monotone when `f` is inducing. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
orderIsoOfIsOpenEmbedding (f : β → α) (h : IsOpenEmbedding f) :
IrreducibleCloseds β ≃o {V : IrreducibleCloseds α | (f ⁻¹' V).Nonempty} | where
toFun T := ⟨map f h.continuous T, nonempty_preimage_closure_image h.continuous T T.2.nonempty⟩
invFun V :=
{ carrier := f ⁻¹' V
isIrreducible' := ⟨V.2, V.1.2.isPreirreducible.preimage h⟩
isClosed' := V.1.3.preimage h.continuous }
left_inv V := by
ext
simp [h.isOpenMap.preimage_closur... | def | TopologicalSpace.IrreducibleCloseds.orderIsoOfIsOpenEmbedding | Topology.Sets | Mathlib/Topology/Sets/Closeds.lean | [] | [
"Set.preimage_mono",
"closure_image_preimage_of_isPreirreducible",
"nonempty_preimage_closure_image"
] | Given `f : U → X` a continuous open embedding, the irreducible closeds of `U` are order isomorphic
to the irreducible closeds of `X` nontrivially intersecting the range of `f`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | IsCompactOpenCovered | Topology.Sets | Mathlib/Topology/Sets/CompactOpenCovered.lean | [] | [
"IsCompact",
"Set",
"TopologicalSpace"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
empty : IsCompactOpenCovered f ∅ | ⟨∅, Set.finite_empty, fun _ _ ↦ ⟨∅, isOpen_empty⟩, fun _ _ ↦ isCompact_empty, by simp⟩ | lemma | IsCompactOpenCovered.empty | Topology.Sets | Mathlib/Topology/Sets/CompactOpenCovered.lean | [] | [
"IsCompactOpenCovered",
"Set.finite_empty",
"isCompact_empty"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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⟩ ↦ ?_⟩
· cases s.eq_empty_or_singleton_of_unique <;> aesop
· refine ⟨{default}, Set.finite_singleton _, fun i h ↦ h ▸ V, fun i ↦ ?_, by simpa⟩
rintro rfl
simpa | lemma | IsCompactOpenCovered.iff_of_unique | Topology.Sets | Mathlib/Topology/Sets/CompactOpenCovered.lean | [] | [
"IsCompact",
"IsCompactOpenCovered",
"Set.finite_singleton",
"Unique"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsCompactOpenCovered.id_iff_isOpen_and_isCompact | Topology.Sets | Mathlib/Topology/Sets/CompactOpenCovered.lean | [] | [
"IsCompact",
"IsCompactOpenCovered",
"IsOpen",
"Set.image_id'",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 ∅, isOpen_sigma_iff.mpr ?_⟩, ?_, ?_⟩
· intro i
by_cases h : i ∈ s
· simpa [h] using (V _ _).2
· simp [h]
· dsimp only
ex... | lemma | IsCompactOpenCovered.iff_isCompactOpenCovered_sigmaMk | Topology.Sets | Mathlib/Topology/Sets/CompactOpenCovered.lean | [] | [
"IsCompactOpenCovered",
"IsOpen",
"Set.image_sigma_eq_iUnion",
"Set.isCompact_sigma",
"Set.mk_preimage_sigma",
"by_cases"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
of_iUnion_eq_of_finite {κ : Type*} [Finite κ] (s : κ → Set S) (hs : ⋃ i, s i = U)
(H : ∀ i, IsCompactOpenCovered f (s i)) : IsCompactOpenCovered f U | by
rw [iff_isCompactOpenCovered_sigmaMk, iff_of_unique]
have (i : κ) : ∃ (V : Opens (Σ i, X i)), IsCompact V.1 ∧ (f _ ·.snd) '' V.1 = s i := by
convert! H i; rw [iff_isCompactOpenCovered_sigmaMk, iff_of_unique]
choose V hVeq hVc using this
exact ⟨⨆ i, V i, by simpa using isCompact_iUnion hVeq, by simp_all [... | lemma | IsCompactOpenCovered.of_iUnion_eq_of_finite | Topology.Sets | Mathlib/Topology/Sets/CompactOpenCovered.lean | [] | [
"Finite",
"IsCompact",
"IsCompactOpenCovered",
"Set",
"Set.image_iUnion",
"isCompact_iUnion"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
of_biUnion_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
have := hf.to_subtype
exact of_iUnion_eq_of_finite (fun i : s ↦ i.1) (by simpa) (by simpa) | lemma | IsCompactOpenCovered.of_biUnion_eq_of_finite | Topology.Sets | Mathlib/Topology/Sets/CompactOpenCovered.lean | [] | [
"IsCompactOpenCovered",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
of_biUnion_eq_of_isCompact [TopologicalSpace S] {U : Set S} (hU : IsCompact U)
(s : Set (Opens S)) (hs : ⋃ t ∈ s, t = U) (H : ∀ t ∈ s, IsCompactOpenCovered f t) :
IsCompactOpenCovered f U | by
classical
obtain ⟨t, ht⟩ := hU.elim_finite_subcover (fun V : s ↦ V.1) (fun V ↦ V.1.2) (by simp [← hs])
refine of_biUnion_eq_of_finite (SetLike.coe '' (t.image Subtype.val : Set (Opens S))) ?_ ?_ ?_
· exact subset_antisymm (fun x h ↦ by aesop) (subset_trans ht <| by simp)
· exact Set.toFinite _
· grind | lemma | IsCompactOpenCovered.of_biUnion_eq_of_isCompact | Topology.Sets | Mathlib/Topology/Sets/CompactOpenCovered.lean | [] | [
"IsCompact",
"IsCompactOpenCovered",
"Set",
"Set.toFinite",
"TopologicalSpace",
"subset_antisymm",
"subset_trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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