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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nhds_eq_pure {a : α} : 𝓝 a = pure a ↔ ¬ IsSuccLimit a | (isOpen_singleton_iff_nhds_eq_pure _).symm.trans isOpen_singleton_iff | theorem | SuccOrder.nhds_eq_pure | Topology.Order | Mathlib/Topology/Order/SuccPred.lean | [] | [
"isOpen_singleton_iff_nhds_eq_pure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_of_isMin {a : α} (h : IsMin a) : 𝓝 a = pure a | by
rw [nhds_eq_pure, isSuccLimit_iff]
tauto | theorem | SuccOrder.nhds_of_isMin | Topology.Order | Mathlib/Topology/Order/SuccPred.lean | [] | [
"IsMin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_bot [OrderBot α] : 𝓝 (⊥ : α) = pure ⊥ | nhds_of_isMin isMin_bot | theorem | SuccOrder.nhds_bot | Topology.Order | Mathlib/Topology/Order/SuccPred.lean | [] | [
"OrderBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_iff {s : Set α} : IsOpen s ↔
∀ o ∈ s, IsSuccLimit o → ∃ a < o, Ioo a o ⊆ s | by
refine isOpen_iff_mem_nhds.trans <| forall₂_congr fun o ho ↦ ?_
by_cases ho' : IsSuccLimit o
· rw [(hasBasis_nhds_Ioc_of_exists_lt (not_isMin_iff.1 ho'.not_isMin)).mem_iff]
grind
· simp [nhds_eq_pure.2 ho', ho, ho'] | theorem | SuccOrder.isOpen_iff | Topology.Order | Mathlib/Topology/Order/SuccPred.lean | [] | [
"IsOpen",
"Set",
"by_cases"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
accPt_principal {a : α} {s : Set α} :
AccPt a (𝓟 s) ↔ ¬ IsMin a ∧ ∀ b < a, (s ∩ Ioo b a).Nonempty | by
rw [accPt_iff_frequently]
by_cases ha : IsMin a
· simp [nhds_of_isMin, ha]
· rw [not_isMin_iff] at ha ⊢
simp_rw [(hasBasis_nhds_Ioc_of_exists_lt ha).frequently_iff, Set.Nonempty, mem_inter_iff]
grind | theorem | SuccOrder.accPt_principal | Topology.Order | Mathlib/Topology/Order/SuccPred.lean | [] | [
"AccPt",
"IsMin",
"Set",
"Set.Nonempty",
"accPt_iff_frequently",
"by_cases",
"not_isMin_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.AccPt.not_isMin {a : α} {s : Set α} (h : AccPt a (𝓟 s)) : ¬ IsMin a | (accPt_principal.1 h).1 | theorem | AccPt.not_isMin | Topology.Order | Mathlib/Topology/Order/SuccPred.lean | [] | [
"AccPt",
"IsMin",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.AccPt.isSuccLimit {a : α} {s : Set α} (h : AccPt a (𝓟 s)) : IsSuccLimit a | by
rw [isSuccLimit_iff, IsSuccPrelimit]
simp_rw [accPt_principal, Set.Nonempty] at h
grind [covBy_iff_Ioo_eq] | theorem | AccPt.isSuccLimit | Topology.Order | Mathlib/Topology/Order/SuccPred.lean | [] | [
"AccPt",
"Set",
"Set.Nonempty",
"covBy_iff_Ioo_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isSuccLimit_of_mem_frontier {a : α} {s : Set α} (ha : a ∈ frontier s) : IsSuccLimit a | by
rw [← isOpen_singleton_iff.not_left]
rw [frontier_eq_closure_inter_closure] at ha
grind [mem_closure_iff, Set.Nonempty] | theorem | SuccOrder.isSuccLimit_of_mem_frontier | Topology.Order | Mathlib/Topology/Order/SuccPred.lean | [] | [
"Set",
"Set.Nonempty",
"frontier",
"frontier_eq_closure_inter_closure",
"mem_closure_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ordConnectedComponent_mem_nhds : ordConnectedComponent s a ∈ 𝓝 a ↔ s ∈ 𝓝 a | by
refine ⟨fun h => mem_of_superset h ordConnectedComponent_subset, fun h => ?_⟩
rcases exists_Icc_mem_subset_of_mem_nhds h with ⟨b, c, ha, ha', hs⟩
exact mem_of_superset ha' (subset_ordConnectedComponent ha hs) | theorem | Set.ordConnectedComponent_mem_nhds | Topology.Order | Mathlib/Topology/Order/T5.lean | [] | [
"exists_Icc_mem_subset_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compl_ordConnectedSection_ordSeparatingSet_mem_nhdsGE (hd : Disjoint s (closure t))
(ha : a ∈ s) : (ordConnectedSection (ordSeparatingSet s t))ᶜ ∈ 𝓝[≥] a | by
have hmem : tᶜ ∈ 𝓝[≥] a := by
refine mem_nhdsWithin_of_mem_nhds ?_
rw [← mem_interior_iff_mem_nhds, interior_compl]
exact disjoint_left.1 hd ha
rcases exists_Icc_mem_subset_of_mem_nhdsGE hmem with ⟨b, hab, hmem', hsub⟩
by_cases H : Disjoint (Icc a b) (ordConnectedSection <| ordSeparatingSet s t)
... | theorem | Set.compl_ordConnectedSection_ordSeparatingSet_mem_nhdsGE | Topology.Order | Mathlib/Topology/Order/T5.lean | [] | [
"Disjoint",
"Set.disjoint_left",
"by_cases",
"closure",
"exists_Icc_mem_subset_of_mem_nhdsGE",
"interior_compl",
"lt_of_not_ge",
"mem_interior_iff_mem_nhds",
"mem_nhdsWithin_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compl_ordConnectedSection_ordSeparatingSet_mem_nhdsLE (hd : Disjoint s (closure t))
(ha : a ∈ s) : (ordConnectedSection <| ordSeparatingSet s t)ᶜ ∈ 𝓝[≤] a | by
have hd' : Disjoint (ofDual ⁻¹' s) (closure <| ofDual ⁻¹' t) := hd
have ha' : toDual a ∈ ofDual ⁻¹' s := ha
simpa only [dual_ordSeparatingSet, dual_ordConnectedSection] using!
compl_ordConnectedSection_ordSeparatingSet_mem_nhdsGE hd' ha' | theorem | Set.compl_ordConnectedSection_ordSeparatingSet_mem_nhdsLE | Topology.Order | Mathlib/Topology/Order/T5.lean | [] | [
"Disjoint",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compl_ordConnectedSection_ordSeparatingSet_mem_nhds (hd : Disjoint s (closure t))
(ha : a ∈ s) : (ordConnectedSection <| ordSeparatingSet s t)ᶜ ∈ 𝓝 a | by
rw [← nhdsLE_sup_nhdsGE, mem_sup]
exact ⟨compl_ordConnectedSection_ordSeparatingSet_mem_nhdsLE hd ha,
compl_ordConnectedSection_ordSeparatingSet_mem_nhdsGE hd ha⟩ | theorem | Set.compl_ordConnectedSection_ordSeparatingSet_mem_nhds | Topology.Order | Mathlib/Topology/Order/T5.lean | [] | [
"Disjoint",
"closure",
"nhdsLE_sup_nhdsGE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ordT5Nhd_mem_nhdsSet (hd : Disjoint s (closure t)) : ordT5Nhd s t ∈ 𝓝ˢ s | bUnion_mem_nhdsSet fun x hx => ordConnectedComponent_mem_nhds.2 <| inter_mem
(by
rw [← mem_interior_iff_mem_nhds, interior_compl]
exact disjoint_left.1 hd hx)
(compl_ordConnectedSection_ordSeparatingSet_mem_nhds hd hx) | theorem | Set.ordT5Nhd_mem_nhdsSet | Topology.Order | Mathlib/Topology/Order/T5.lean | [] | [
"Disjoint",
"bUnion_mem_nhdsSet",
"closure",
"interior_compl",
"mem_interior_iff_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSet (α : Type*) [Preorder α] : TopologicalSpace α | where
IsOpen := IsUpperSet
isOpen_univ := isUpperSet_univ
isOpen_inter _ _ := IsUpperSet.inter
isOpen_sUnion _ := isUpperSet_sUnion | def | Topology.upperSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"IsOpen",
"IsUpperSet",
"IsUpperSet.inter",
"Preorder",
"TopologicalSpace",
"isOpen_sUnion",
"isOpen_univ",
"isUpperSet_sUnion",
"isUpperSet_univ"
] | Topology whose open sets are upper sets.
Note: In general the upper set topology does not coincide with the upper topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lowerSet (α : Type*) [Preorder α] : TopologicalSpace α | where
IsOpen := IsLowerSet
isOpen_univ := isLowerSet_univ
isOpen_inter _ _ := IsLowerSet.inter
isOpen_sUnion _ := isLowerSet_sUnion | def | Topology.lowerSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"IsOpen",
"Preorder",
"TopologicalSpace",
"isOpen_sUnion",
"isOpen_univ"
] | Topology whose open sets are lower sets.
Note: In general the lower set topology does not coincide with the lower topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
WithUpperSet (α : Type*) | α | def | Topology.WithUpperSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | Type synonym for a preorder equipped with the upper set topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toUpperSet : α ≃ WithUpperSet α | Equiv.refl _ | def | Topology.WithUpperSet.toUpperSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Equiv.refl"
] | `toUpperSet` is the identity function to the `WithUpperSet` of a type. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ofUpperSet : WithUpperSet α ≃ α | Equiv.refl _ | def | Topology.WithUpperSet.ofUpperSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Equiv.refl"
] | `ofUpperSet` is the identity function from the `WithUpperSet` of a type. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toUpperSet_symm : (@toUpperSet α).symm = ofUpperSet | rfl | lemma | Topology.WithUpperSet.toUpperSet_symm | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofUpperSet_symm : (@ofUpperSet α).symm = toUpperSet | rfl | lemma | Topology.WithUpperSet.ofUpperSet_symm | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toUpperSet_ofUpperSet (a : WithUpperSet α) : toUpperSet (ofUpperSet a) = a | rfl | lemma | Topology.WithUpperSet.toUpperSet_ofUpperSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofUpperSet_toUpperSet (a : α) : ofUpperSet (toUpperSet a) = a | rfl | lemma | Topology.WithUpperSet.ofUpperSet_toUpperSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toUpperSet_inj {a b : α} : toUpperSet a = toUpperSet b ↔ a = b | Iff.rfl | lemma | Topology.WithUpperSet.toUpperSet_inj | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofUpperSet_inj {a b : WithUpperSet α} : ofUpperSet a = ofUpperSet b ↔ a = b | Iff.rfl | lemma | Topology.WithUpperSet.ofUpperSet_inj | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rec {motive : WithUpperSet α → Sort*} (toUpperSet : ∀ a, motive (toUpperSet a)) :
∀ a, motive a | fun a => toUpperSet (ofUpperSet a) | def | Topology.WithUpperSet.rec | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | A recursor for `WithUpperSet`. Use as `induction x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ofUpperSet_le_iff {a b : WithUpperSet α} : ofUpperSet a ≤ ofUpperSet b ↔ a ≤ b | Iff.rfl | lemma | Topology.WithUpperSet.ofUpperSet_le_iff | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toUpperSet_le_iff {a b : α} : toUpperSet a ≤ toUpperSet b ↔ a ≤ b | Iff.rfl | lemma | Topology.WithUpperSet.toUpperSet_le_iff | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofUpperSetOrderIso : WithUpperSet α ≃o α | where
toEquiv := ofUpperSet
map_rel_iff' := ofUpperSet_le_iff | def | Topology.WithUpperSet.ofUpperSetOrderIso | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | `ofUpperSet` as an `OrderIso` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toUpperSetOrderIso : α ≃o WithUpperSet α | where
toEquiv := toUpperSet
map_rel_iff' := toUpperSet_le_iff | def | Topology.WithUpperSet.toUpperSetOrderIso | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | `toUpperSet` as an `OrderIso` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
WithLowerSet (α : Type*) | α | def | Topology.WithLowerSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | Type synonym for a preorder equipped with the lower set topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toLowerSet : α ≃ WithLowerSet α | Equiv.refl _ | def | Topology.WithLowerSet.toLowerSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Equiv.refl"
] | `toLowerSet` is the identity function to the `WithLowerSet` of a type. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ofLowerSet : WithLowerSet α ≃ α | Equiv.refl _ | def | Topology.WithLowerSet.ofLowerSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Equiv.refl"
] | `ofLowerSet` is the identity function from the `WithLowerSet` of a type. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toLowerSet_symm : (@toLowerSet α).symm = ofLowerSet | rfl | lemma | Topology.WithLowerSet.toLowerSet_symm | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofLowerSet_symm : (@ofLowerSet α).symm = toLowerSet | rfl | lemma | Topology.WithLowerSet.ofLowerSet_symm | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toLowerSet_ofLowerSet (a : WithLowerSet α) : toLowerSet (ofLowerSet a) = a | rfl | lemma | Topology.WithLowerSet.toLowerSet_ofLowerSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofLowerSet_toLowerSet (a : α) : ofLowerSet (toLowerSet a) = a | rfl | lemma | Topology.WithLowerSet.ofLowerSet_toLowerSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toLowerSet_inj {a b : α} : toLowerSet a = toLowerSet b ↔ a = b | Iff.rfl | lemma | Topology.WithLowerSet.toLowerSet_inj | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofLowerSet_inj {a b : WithLowerSet α} : ofLowerSet a = ofLowerSet b ↔ a = b | Iff.rfl | lemma | Topology.WithLowerSet.ofLowerSet_inj | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rec {motive : WithLowerSet α → Sort*} (toLowerSet : ∀ a, motive (toLowerSet a)) :
∀ a, motive a | fun a => toLowerSet (ofLowerSet a) | def | Topology.WithLowerSet.rec | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | A recursor for `WithLowerSet`. Use as `induction x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ofLowerSet_le_iff {a b : WithLowerSet α} : ofLowerSet a ≤ ofLowerSet b ↔ a ≤ b | Iff.rfl | lemma | Topology.WithLowerSet.ofLowerSet_le_iff | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toLowerSet_le_iff {a b : α} : toLowerSet a ≤ toLowerSet b ↔ a ≤ b | Iff.rfl | lemma | Topology.WithLowerSet.toLowerSet_le_iff | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofLowerSetOrderIso : WithLowerSet α ≃o α | where
toEquiv := ofLowerSet
map_rel_iff' := ofLowerSet_le_iff | def | Topology.WithLowerSet.ofLowerSetOrderIso | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | `ofLowerSet` as an `OrderIso` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toLowerSetOrderIso : α ≃o WithLowerSet α | where
toEquiv := toLowerSet
map_rel_iff' := toLowerSet_le_iff | def | Topology.WithLowerSet.toLowerSetOrderIso | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | `toLowerSet` as an `OrderIso` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
WithUpperSet.toDualHomeomorph [Preorder α] : WithUpperSet α ≃ₜ WithLowerSet αᵒᵈ | where
toFun := OrderDual.toDual
invFun := OrderDual.ofDual
left_inv := OrderDual.toDual_ofDual
right_inv := OrderDual.ofDual_toDual
continuous_toFun := continuous_coinduced_rng
continuous_invFun := continuous_coinduced_rng | def | Topology.WithUpperSet.toDualHomeomorph | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"OrderDual.ofDual",
"OrderDual.ofDual_toDual",
"OrderDual.toDual",
"OrderDual.toDual_ofDual",
"Preorder",
"continuous_coinduced_rng"
] | The Upper Set topology is homeomorphic to the Lower Set topology on the dual order | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsUpperSet (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
topology_eq_upperSetTopology : t = upperSet α | class | Topology.IsUpperSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"IsUpperSet",
"Preorder",
"TopologicalSpace"
] | Prop-valued mixin for an ordered topological space to be
The upper set topology is the topology where the open sets are the upper sets. In general the upper
set topology does not coincide with the upper topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLowerSet (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
topology_eq_lowerSetTopology : t = lowerSet α | class | Topology.IsLowerSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Preorder",
"TopologicalSpace"
] | The lower set topology is the topology where the open sets are the lower sets. In general the lower
set topology does not coincide with the lower topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
topology_eq : ‹_› = upperSet α | topology_eq_upperSetTopology | lemma | Topology.IsUpperSet.topology_eq | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.OrderDual.instIsLowerSet : Topology.IsLowerSet αᵒᵈ | where
topology_eq_lowerSetTopology := by ext; rw [IsUpperSet.topology_eq α] | instance | OrderDual.instIsLowerSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Topology.IsLowerSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
WithUpperSetHomeomorph : WithUpperSet α ≃ₜ α | WithUpperSet.ofUpperSet.toHomeomorphOfIsInducing ⟨topology_eq α ▸ induced_id.symm⟩ | def | Topology.IsUpperSet.WithUpperSetHomeomorph | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | If `α` is equipped with the upper set topology, then it is homeomorphic to
`WithUpperSet α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_iff_isUpperSet : IsOpen s ↔ IsUpperSet s | by
rw [topology_eq α]
rfl | lemma | Topology.IsUpperSet.isOpen_iff_isUpperSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"IsOpen",
"IsUpperSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toAlexandrovDiscrete : AlexandrovDiscrete α | where
isOpen_sInter S := by simpa only [isOpen_iff_isUpperSet] using isUpperSet_sInter (α := α) | instance | Topology.IsUpperSet.toAlexandrovDiscrete | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"AlexandrovDiscrete",
"isOpen_sInter",
"isUpperSet_sInter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_iff_isLower : IsClosed s ↔ IsLowerSet s | by
rw [← isOpen_compl_iff, isOpen_iff_isUpperSet,
isLowerSet_compl.symm, compl_compl] | lemma | Topology.IsUpperSet.isClosed_iff_isLower | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"IsClosed",
"compl_compl",
"isOpen_compl_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_eq_lowerClosure {s : Set α} : closure s = lowerClosure s | by
rw [subset_antisymm_iff]
refine ⟨?_, lowerClosure_min subset_closure (isClosed_iff_isLower.1 isClosed_closure)⟩
· apply closure_minimal subset_lowerClosure _
rw [isClosed_iff_isLower]
exact LowerSet.lower (lowerClosure s) | lemma | Topology.IsUpperSet.closure_eq_lowerClosure | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Set",
"closure",
"closure_minimal",
"isClosed_closure",
"subset_antisymm_iff",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_singleton {a : α} : closure {a} = Iic a | by
rw [closure_eq_lowerClosure, lowerClosure_singleton]
rfl | lemma | Topology.IsUpperSet.closure_singleton | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"closure",
"closure_singleton"
] | The closure of a singleton `{a}` in the upper set topology is the right-closed left-infinite
interval $(-∞,a]$. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
specializes_iff_le {a b : α} : a ⤳ b ↔ b ≤ a | by
simp only [specializes_iff_closure_subset, closure_singleton, Iic_subset_Iic] | lemma | Topology.IsUpperSet.specializes_iff_le | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"closure_singleton",
"specializes_iff_closure_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsKer_eq_upperClosure (s : Set α) : nhdsKer s = ↑(upperClosure s) | by
ext; simp [mem_nhdsKer_iff_specializes, specializes_iff_le] | lemma | Topology.IsUpperSet.nhdsKer_eq_upperClosure | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Set",
"mem_nhdsKer_iff_specializes",
"nhdsKer",
"upperClosure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsKer_singleton (a : α) : nhdsKer {a} = Ici a | by
rw [nhdsKer_eq_upperClosure, upperClosure_singleton, UpperSet.coe_Ici] | lemma | Topology.IsUpperSet.nhdsKer_singleton | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"UpperSet.coe_Ici",
"nhdsKer",
"upperClosure_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_eq_principal_Ici (a : α) : 𝓝 a = 𝓟 (Ici a) | by
rw [← principal_nhdsKer_singleton, nhdsKer_singleton] | lemma | Topology.IsUpperSet.nhds_eq_principal_Ici | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"principal_nhdsKer_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsSet_eq_principal_upperClosure (s : Set α) : 𝓝ˢ s = 𝓟 ↑(upperClosure s) | by
rw [← principal_nhdsKer, nhdsKer_eq_upperClosure] | lemma | Topology.IsUpperSet.nhdsSet_eq_principal_upperClosure | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Set",
"principal_nhdsKer",
"upperClosure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Topology.isUpperSet_iff_nhds {α : Type*} [TopologicalSpace α] [Preorder α] :
Topology.IsUpperSet α ↔ (∀ a : α, 𝓝 a = 𝓟 (Ici a)) | where
mp _ a := nhds_eq_principal_Ici a
mpr hα := ⟨by simp [TopologicalSpace.ext_iff_nhds, hα, nhds_eq_principal_Ici]⟩ | lemma | Topology.isUpperSet_iff_nhds | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Preorder",
"TopologicalSpace",
"TopologicalSpace.ext_iff_nhds",
"Topology.IsUpperSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f | by
constructor
· intro hf
simp_rw [continuous_def, isOpen_iff_isUpperSet]
exact fun _ hs ↦ IsUpperSet.preimage hs hf
· intro hf a b hab
rw [← mem_Iic, ← closure_singleton] at hab ⊢
apply Continuous.closure_preimage_subset hf {f b}
apply mem_of_mem_of_subset hab
apply closure_mono
rw [s... | lemma | Topology.IsUpperSet.monotone_iff_continuous | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Continuous",
"Continuous.closure_preimage_subset",
"IsUpperSet.preimage",
"Monotone",
"TopologicalSpace",
"Topology.IsUpperSet",
"closure_mono",
"closure_singleton",
"continuous_def"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
monotone_to_upperTopology_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [IsUpper β] {f : α → β} (hf : Monotone f) : Continuous f | by
simp_rw [continuous_def, isOpen_iff_isUpperSet]
intro s hs
exact (IsUpper.isUpperSet_of_isOpen hs).preimage hf | lemma | Topology.IsUpperSet.monotone_to_upperTopology_continuous | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Continuous",
"Monotone",
"TopologicalSpace",
"Topology.IsUpperSet",
"continuous_def"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperSet_le_upper {t₁ t₂ : TopologicalSpace α} [@Topology.IsUpperSet α t₁ _]
[@Topology.IsUpper α t₂ _] : t₁ ≤ t₂ | fun s hs => by
rw [@isOpen_iff_isUpperSet α _ t₁]
exact IsUpper.isUpperSet_of_isOpen hs | lemma | Topology.IsUpperSet.upperSet_le_upper | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"TopologicalSpace",
"Topology.IsUpper",
"Topology.IsUpperSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
topology_eq : ‹_› = lowerSet α | topology_eq_lowerSetTopology | lemma | Topology.IsLowerSet.topology_eq | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.OrderDual.instIsUpperSet : Topology.IsUpperSet αᵒᵈ | where
topology_eq_upperSetTopology := by ext; rw [IsLowerSet.topology_eq α] | instance | OrderDual.instIsUpperSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Topology.IsUpperSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
WithLowerSetHomeomorph : WithLowerSet α ≃ₜ α | WithLowerSet.ofLowerSet.toHomeomorphOfIsInducing ⟨topology_eq α ▸ induced_id.symm⟩ | def | Topology.IsLowerSet.WithLowerSetHomeomorph | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | If `α` is equipped with the lower set topology, then it is homeomorphic to `WithLowerSet α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_iff_isLowerSet : IsOpen s ↔ IsLowerSet s | by rw [topology_eq α]; rfl | lemma | Topology.IsLowerSet.isOpen_iff_isLowerSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"IsOpen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toAlexandrovDiscrete : AlexandrovDiscrete α | IsUpperSet.toAlexandrovDiscrete (α := αᵒᵈ) | instance | Topology.IsLowerSet.toAlexandrovDiscrete | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"AlexandrovDiscrete"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_iff_isUpper : IsClosed s ↔ IsUpperSet s | by
rw [← isOpen_compl_iff, isOpen_iff_isLowerSet, isUpperSet_compl.symm, compl_compl] | lemma | Topology.IsLowerSet.isClosed_iff_isUpper | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"IsClosed",
"IsUpperSet",
"compl_compl",
"isOpen_compl_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_eq_upperClosure {s : Set α} : closure s = upperClosure s | IsUpperSet.closure_eq_lowerClosure (α := αᵒᵈ) | lemma | Topology.IsLowerSet.closure_eq_upperClosure | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Set",
"closure",
"upperClosure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_singleton {a : α} : closure {a} = Ici a | by
rw [closure_eq_upperClosure, upperClosure_singleton]
rfl | lemma | Topology.IsLowerSet.closure_singleton | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"closure",
"closure_singleton",
"upperClosure_singleton"
] | The closure of a singleton `{a}` in the lower set topology is the right-closed left-infinite
interval $(-∞,a]$. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
specializes_iff_le {a b : α} : a ⤳ b ↔ a ≤ b | by
simp only [specializes_iff_closure_subset, closure_singleton, Ici_subset_Ici] | lemma | Topology.IsLowerSet.specializes_iff_le | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"closure_singleton",
"specializes_iff_closure_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsKer_eq_lowerClosure (s : Set α) : nhdsKer s = ↑(lowerClosure s) | by
ext; simp [mem_nhdsKer_iff_specializes, specializes_iff_le] | lemma | Topology.IsLowerSet.nhdsKer_eq_lowerClosure | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Set",
"mem_nhdsKer_iff_specializes",
"nhdsKer"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsKer_singleton (a : α) : nhdsKer {a} = Iic a | by
rw [nhdsKer_eq_lowerClosure, lowerClosure_singleton, LowerSet.coe_Iic] | lemma | Topology.IsLowerSet.nhdsKer_singleton | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"nhdsKer"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_eq_principal_Iic (a : α) : 𝓝 a = 𝓟 (Iic a) | by
rw [← principal_nhdsKer_singleton, nhdsKer_singleton] | lemma | Topology.IsLowerSet.nhds_eq_principal_Iic | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"principal_nhdsKer_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsSet_eq_principal_lowerClosure (s : Set α) : 𝓝ˢ s = 𝓟 ↑(lowerClosure s) | by
rw [← principal_nhdsKer, nhdsKer_eq_lowerClosure] | lemma | Topology.IsLowerSet.nhdsSet_eq_principal_lowerClosure | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Set",
"principal_nhdsKer"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Topology.isLowerSet_iff_nhds {α : Type*} [TopologicalSpace α] [Preorder α] :
Topology.IsLowerSet α ↔ (∀ a : α, 𝓝 a = 𝓟 (Iic a)) | where
mp _ a := nhds_eq_principal_Iic a
mpr hα := ⟨by simp [TopologicalSpace.ext_iff_nhds, hα, nhds_eq_principal_Iic]⟩ | lemma | Topology.isLowerSet_iff_nhds | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Preorder",
"TopologicalSpace",
"TopologicalSpace.ext_iff_nhds",
"Topology.IsLowerSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsLowerSet α] [Topology.IsLowerSet β] {f : α → β} : Monotone f ↔ Continuous f | by
rw [← monotone_dual_iff]
exact IsUpperSet.monotone_iff_continuous (α := αᵒᵈ) (β := βᵒᵈ)
(f := (toDual ∘ f ∘ ofDual : αᵒᵈ → βᵒᵈ)) | lemma | Topology.IsLowerSet.monotone_iff_continuous | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Continuous",
"Monotone",
"TopologicalSpace",
"Topology.IsLowerSet",
"monotone_dual_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
monotone_to_lowerTopology_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsLowerSet α] [IsLower β] {f : α → β} (hf : Monotone f) : Continuous f | IsUpperSet.monotone_to_upperTopology_continuous (α := αᵒᵈ) (β := βᵒᵈ) hf.dual | lemma | Topology.IsLowerSet.monotone_to_lowerTopology_continuous | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Continuous",
"Monotone",
"TopologicalSpace",
"Topology.IsLowerSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerSet_le_lower {t₁ t₂ : TopologicalSpace α} [@Topology.IsLowerSet α t₁ _]
[@IsLower α t₂ _] : t₁ ≤ t₂ | fun s hs => by
rw [@isOpen_iff_isLowerSet α _ t₁]
exact IsLower.isLowerSet_of_isOpen hs | lemma | Topology.IsLowerSet.lowerSet_le_lower | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"TopologicalSpace",
"Topology.IsLowerSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUpperSet_orderDual [Preorder α] [TopologicalSpace α] :
Topology.IsUpperSet αᵒᵈ ↔ Topology.IsLowerSet α | by
constructor
· apply OrderDual.instIsLowerSet
· apply OrderDual.instIsUpperSet | lemma | Topology.isUpperSet_orderDual | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"OrderDual.instIsLowerSet",
"OrderDual.instIsUpperSet",
"Preorder",
"TopologicalSpace",
"Topology.IsLowerSet",
"Topology.IsUpperSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLowerSet_orderDual [Preorder α] [TopologicalSpace α] :
Topology.IsLowerSet αᵒᵈ ↔ Topology.IsUpperSet α | isUpperSet_orderDual.symm | lemma | Topology.isLowerSet_orderDual | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"Preorder",
"TopologicalSpace",
"Topology.IsLowerSet",
"Topology.IsUpperSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map (f : α →o β) : C(WithUpperSet α, WithUpperSet β) | where
toFun := toUpperSet ∘ f ∘ ofUpperSet
continuous_toFun := continuous_def.2 fun _s hs ↦ IsUpperSet.preimage hs f.monotone | def | Topology.WithUpperSet.map | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"IsUpperSet.preimage"
] | A monotone map between preorders spaces induces a continuous map between themselves considered
with the upper set topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
map_id : map (OrderHom.id : α →o α) = ContinuousMap.id _ | rfl | lemma | Topology.WithUpperSet.map_id | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"ContinuousMap.id",
"OrderHom.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_comp (g : β →o γ) (f : α →o β) : map (g.comp f) = (map g).comp (map f) | rfl | lemma | Topology.WithUpperSet.map_comp | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toUpperSet_specializes_toUpperSet {a b : α} :
toUpperSet a ⤳ toUpperSet b ↔ b ≤ a | by
simp_rw [specializes_iff_closure_subset, IsUpperSet.closure_singleton, Iic_subset_Iic,
toUpperSet_le_iff] | lemma | Topology.WithUpperSet.toUpperSet_specializes_toUpperSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"specializes_iff_closure_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofUpperSet_le_ofUpperSet {a b : WithUpperSet α} :
ofUpperSet a ≤ ofUpperSet b ↔ b ⤳ a | toUpperSet_specializes_toUpperSet.symm | lemma | Topology.WithUpperSet.ofUpperSet_le_ofUpperSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUpperSet_toUpperSet_preimage {s : Set (WithUpperSet α)} :
IsUpperSet (toUpperSet ⁻¹' s) ↔ IsOpen s | Iff.rfl | lemma | Topology.WithUpperSet.isUpperSet_toUpperSet_preimage | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"IsOpen",
"IsUpperSet",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_ofUpperSet_preimage {s : Set α} :
IsOpen (ofUpperSet ⁻¹' s) ↔ IsUpperSet s | isUpperSet_toUpperSet_preimage.symm | lemma | Topology.WithUpperSet.isOpen_ofUpperSet_preimage | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"IsOpen",
"IsUpperSet",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map (f : α →o β) : C(WithLowerSet α, WithLowerSet β) | where
toFun := toLowerSet ∘ f ∘ ofLowerSet
continuous_toFun := continuous_def.2 fun _s hs ↦ IsLowerSet.preimage hs f.monotone | def | Topology.WithLowerSet.map | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | A monotone map between preorders spaces induces a continuous map between themselves considered
with the lower set topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toLowerSet_specializes_toLowerSet {a b : α} :
toLowerSet a ⤳ toLowerSet b ↔ a ≤ b | by
simp_rw [specializes_iff_closure_subset, IsLowerSet.closure_singleton, Ici_subset_Ici,
toLowerSet_le_iff] | lemma | Topology.WithLowerSet.toLowerSet_specializes_toLowerSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"specializes_iff_closure_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofLowerSet_le_ofLowerSet {a b : WithLowerSet α} :
ofLowerSet a ≤ ofLowerSet b ↔ a ⤳ b | toLowerSet_specializes_toLowerSet.symm | lemma | Topology.WithLowerSet.ofLowerSet_le_ofLowerSet | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLowerSet_toLowerSet_preimage {s : Set (WithLowerSet α)} :
IsLowerSet (toLowerSet ⁻¹' s) ↔ IsOpen s | Iff.rfl | lemma | Topology.WithLowerSet.isLowerSet_toLowerSet_preimage | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_ofLowerSet_preimage {s : Set α} :
IsOpen (ofLowerSet ⁻¹' s) ↔ IsLowerSet s | isLowerSet_toLowerSet_preimage.symm | lemma | Topology.WithLowerSet.isOpen_ofLowerSet_preimage | Topology.Order | Mathlib/Topology/Order/UpperLowerSetTopology.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isEmbedding_coe : Topology.IsEmbedding ((↑) : ι → WithTop ι) | by
refine WithTop.coe_strictMono.isEmbedding_of_ordConnected (α := ι) ?_
rw [WithTop.range_coe]
exact Set.ordConnected_Iio | lemma | WithTop.isEmbedding_coe | Topology.Order | Mathlib/Topology/Order/WithTop.lean | [] | [
"Topology.IsEmbedding",
"WithTop.range_coe"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpenEmbedding_coe : Topology.IsOpenEmbedding ((↑) : ι → WithTop ι) | ⟨isEmbedding_coe, by rw [WithTop.range_coe]; exact isOpen_Iio⟩ | lemma | WithTop.isOpenEmbedding_coe | Topology.Order | Mathlib/Topology/Order/WithTop.lean | [] | [
"Topology.IsOpenEmbedding",
"WithTop.range_coe"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_coe {r : ι} : 𝓝 (r : WithTop ι) = (𝓝 r).map (↑) | (isOpenEmbedding_coe.map_nhds_eq r).symm | lemma | WithTop.nhds_coe | Topology.Order | Mathlib/Topology/Order/WithTop.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_coe : Continuous ((↑) : ι → WithTop ι) | isEmbedding_coe.continuous | lemma | WithTop.continuous_coe | Topology.Order | Mathlib/Topology/Order/WithTop.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_untopD (d : ι) {a : WithTop ι} (ha : a ≠ ⊤) :
Tendsto (untopD d) (𝓝 a) (𝓝 (untopD d a)) | by
lift a to ι using ha
rw [nhds_coe, tendsto_map'_iff]
exact tendsto_id | lemma | WithTop.tendsto_untopD | Topology.Order | Mathlib/Topology/Order/WithTop.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousOn_untopD (d : ι) : ContinuousOn (untopD d) { a : WithTop ι | a ≠ ⊤ } | fun _a ha ↦ ContinuousAt.continuousWithinAt (tendsto_untopD d ha) | lemma | WithTop.continuousOn_untopD | Topology.Order | Mathlib/Topology/Order/WithTop.lean | [] | [
"ContinuousAt.continuousWithinAt",
"ContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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