fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
exists_isClopen_lower_of_not_le (h : ¬x ≤ y) :
∃ U : Set α, IsClopen U ∧ IsLowerSet U ∧ x ∉ U ∧ y ∈ U :=
let ⟨U, hU, hU', hx, hy⟩ := exists_isClopen_upper_of_not_le h
⟨Uᶜ, hU.compl, hU'.compl, Classical.not_not.2 hx, hy⟩ | theorem | Topology | [
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/Priestley.lean | exists_isClopen_lower_of_not_le | null |
exists_isClopen_upper_or_lower_of_ne (h : x ≠ y) :
∃ U : Set α, IsClopen U ∧ (IsUpperSet U ∨ IsLowerSet U) ∧ x ∈ U ∧ y ∉ U := by
obtain h | h := h.not_le_or_not_ge
· exact (exists_isClopen_upper_of_not_le h).imp fun _ ↦ And.imp_right <| And.imp_left Or.inl
· obtain ⟨U, hU, hU', hy, hx⟩ := exists_isClopen_lowe... | theorem | Topology | [
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/Priestley.lean | exists_isClopen_upper_or_lower_of_ne | null |
protected Filter.Tendsto.IccExtend (f : γ → Icc a b → β) {la : Filter α} {lb : Filter β}
{lc : Filter γ} (hf : Tendsto ↿f (lc ×ˢ la.map (projIcc a b h)) lb) :
Tendsto (↿(IccExtend h ∘ f)) (lc ×ˢ la) lb :=
hf.comp <| tendsto_id.prodMap tendsto_map
variable [TopologicalSpace α] [OrderTopology α] [TopologicalSpa... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/ProjIcc.lean | Filter.Tendsto.IccExtend | null |
continuous_projIcc : Continuous (projIcc a b h) :=
(continuous_const.max <| continuous_const.min continuous_id).subtype_mk _ | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/ProjIcc.lean | continuous_projIcc | null |
isQuotientMap_projIcc : IsQuotientMap (projIcc a b h) :=
isQuotientMap_iff.2 ⟨projIcc_surjective h, fun s =>
⟨fun hs => hs.preimage continuous_projIcc, fun hs => ⟨_, hs, by ext; simp⟩⟩⟩
@[simp] | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/ProjIcc.lean | isQuotientMap_projIcc | null |
continuous_IccExtend_iff {f : Icc a b → β} : Continuous (IccExtend h f) ↔ Continuous f :=
isQuotientMap_projIcc.continuous_iff.symm | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/ProjIcc.lean | continuous_IccExtend_iff | null |
@[fun_prop]
protected Continuous.IccExtend {f : γ → Icc a b → β} {g : γ → α} (hf : Continuous ↿f)
(hg : Continuous g) : Continuous fun a => IccExtend h (f a) (g a) :=
show Continuous (↿f ∘ fun x => (x, projIcc a b h (g x)))
from hf.comp <| continuous_id.prodMk <| continuous_projIcc.comp hg | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/ProjIcc.lean | Continuous.IccExtend | See Note [continuity lemma statement]. |
@[continuity, fun_prop]
protected Continuous.Icc_extend' {f : Icc a b → β} (hf : Continuous f) :
Continuous (IccExtend h f) :=
hf.comp continuous_projIcc
@[fun_prop] | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/ProjIcc.lean | Continuous.Icc_extend' | A useful special case of `Continuous.IccExtend`. |
ContinuousAt.IccExtend {x : γ} (f : γ → Icc a b → β) {g : γ → α}
(hf : ContinuousAt ↿f (x, projIcc a b h (g x))) (hg : ContinuousAt g x) :
ContinuousAt (fun a => IccExtend h (f a) (g a)) x :=
show ContinuousAt (↿f ∘ fun x => (x, projIcc a b h (g x))) x from
ContinuousAt.comp hf <| continuousAt_id.prodMk <... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/ProjIcc.lean | ContinuousAt.IccExtend | null |
denseRange_ratCast : DenseRange (fun r : ℚ ↦ ((r : ℝ) : EReal)) :=
dense_of_exists_between fun _ _ h => exists_range_iff.2 <| exists_rat_btwn_of_lt h | lemma | Topology | [
"Mathlib.Data.EReal.Basic",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Order/Real.lean | denseRange_ratCast | null |
exists_Ioo_extr_on_Icc (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) :
∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c := by
have ne : (Icc a b).Nonempty := nonempty_Icc.2 (le_of_lt hab)
obtain ⟨c, cmem, cle⟩ : ∃ c ∈ Icc a b, ∀ x ∈ Icc a b, f c ≤ f x :=
isCompact_Icc.exists_isMinOn ne hfc
obtain ... | theorem | Topology | [
"Mathlib.Topology.Order.ExtendFrom",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Order/Rolle.lean | exists_Ioo_extr_on_Icc | A continuous function on a closed interval with `f a = f b`
takes either its maximum or its minimum value at a point in the interior of the interval. |
exists_isLocalExtr_Ioo (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) :
∃ c ∈ Ioo a b, IsLocalExtr f c :=
let ⟨c, cmem, hc⟩ := exists_Ioo_extr_on_Icc hab hfc hfI
⟨c, cmem, hc.isLocalExtr <| Icc_mem_nhds cmem.1 cmem.2⟩ | theorem | Topology | [
"Mathlib.Topology.Order.ExtendFrom",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Order/Rolle.lean | exists_isLocalExtr_Ioo | A continuous function on a closed interval with `f a = f b`
has a local extremum at some point of the corresponding open interval. |
exists_isExtrOn_Ioo_of_tendsto (hab : a < b) (hfc : ContinuousOn f (Ioo a b))
(ha : Tendsto f (𝓝[>] a) (𝓝 l)) (hb : Tendsto f (𝓝[<] b) (𝓝 l)) :
∃ c ∈ Ioo a b, IsExtrOn f (Ioo a b) c := by
have h : EqOn (extendFrom (Ioo a b) f) f (Ioo a b) := extendFrom_extends hfc
obtain ⟨c, hc, hfc⟩ : ∃ c ∈ Ioo a b, Is... | lemma | Topology | [
"Mathlib.Topology.Order.ExtendFrom",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Order/Rolle.lean | exists_isExtrOn_Ioo_of_tendsto | If a function `f` is continuous on an open interval
and tends to the same value at its endpoints, then it has an extremum on this open interval. |
exists_isLocalExtr_Ioo_of_tendsto (hab : a < b) (hfc : ContinuousOn f (Ioo a b))
(ha : Tendsto f (𝓝[>] a) (𝓝 l)) (hb : Tendsto f (𝓝[<] b) (𝓝 l)) :
∃ c ∈ Ioo a b, IsLocalExtr f c :=
let ⟨c, cmem, hc⟩ := exists_isExtrOn_Ioo_of_tendsto hab hfc ha hb
⟨c, cmem, hc.isLocalExtr <| Ioo_mem_nhds cmem.1 cmem.2⟩ | lemma | Topology | [
"Mathlib.Topology.Order.ExtendFrom",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Order/Rolle.lean | exists_isLocalExtr_Ioo_of_tendsto | If a function `f` is continuous on an open interval
and tends to the same value at its endpoints,
then it has a local extremum on this open interval. |
of `IsScott`.
A class `Scott` is defined in `Topology/OmegaCompletePartialOrder` and made an instance of a
topological space by defining the open sets to be those which have characteristic functions which
are monotone and preserve limits of countable chains (`OmegaCompletePartialOrder.Continuous'`).
A Scott continuous ... | instance | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | of | null |
DirSupInaccOn (D : Set (Set α)) (s : Set α) : Prop :=
∀ ⦃d⦄, d ∈ D → d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a⦄, IsLUB d a → a ∈ s → (d ∩ s).Nonempty | def | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | DirSupInaccOn | A set `s` is said to be inaccessible by directed joins on `D` if, when the least upper bound of
a directed set `d` in `D` lies in `s` then `d` has non-empty intersection with `s`. |
DirSupInacc (s : Set α) : Prop :=
∀ ⦃d⦄, d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a⦄, IsLUB d a → a ∈ s → (d ∩ s).Nonempty
@[simp] lemma dirSupInaccOn_univ : DirSupInaccOn univ s ↔ DirSupInacc s := by
simp [DirSupInaccOn, DirSupInacc]
@[simp] lemma DirSupInacc.dirSupInaccOn {D : Set (Set α)} :
DirSupInacc s → Dir... | def | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | DirSupInacc | A set `s` is said to be inaccessible by directed joins if, when the least upper bound of a
directed set `d` lies in `s` then `d` has non-empty intersection with `s`. |
DirSupInaccOn.mono {D₁ D₂ : Set (Set α)} (hD : D₁ ⊆ D₂) (hf : DirSupInaccOn D₂ s) :
DirSupInaccOn D₁ s := fun ⦃_⦄ a ↦ hf (hD a) | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | DirSupInaccOn.mono | null |
DirSupClosed (s : Set α) : Prop :=
∀ ⦃d⦄, d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a⦄, IsLUB d a → d ⊆ s → a ∈ s
@[simp] lemma dirSupInacc_compl : DirSupInacc sᶜ ↔ DirSupClosed s := by
simp [DirSupInacc, DirSupClosed, ← not_disjoint_iff_nonempty_inter, not_imp_not,
disjoint_compl_right_iff]
@[simp] lemma dirSupCl... | def | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | DirSupClosed | A set `s` is said to be closed under directed joins if, whenever a directed set `d` has a least
upper bound `a` and is a subset of `s` then `a` also lies in `s`. |
DirSupClosed.inter (hs : DirSupClosed s) (ht : DirSupClosed t) : DirSupClosed (s ∩ t) :=
fun _d hd hd' _a ha hds ↦ ⟨hs hd hd' ha <| hds.trans inter_subset_left,
ht hd hd' ha <| hds.trans inter_subset_right⟩ | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | DirSupClosed.inter | null |
DirSupInacc.union (hs : DirSupInacc s) (ht : DirSupInacc t) : DirSupInacc (s ∪ t) := by
rw [← dirSupClosed_compl, compl_union]; exact hs.compl.inter ht.compl | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | DirSupInacc.union | null |
IsUpperSet.dirSupClosed (hs : IsUpperSet s) : DirSupClosed s :=
fun _d ⟨_b, hb⟩ _ _a ha hds ↦ hs (ha.1 hb) <| hds hb | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | IsUpperSet.dirSupClosed | null |
IsLowerSet.dirSupInacc (hs : IsLowerSet s) : DirSupInacc s := hs.compl.dirSupClosed.of_compl | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | IsLowerSet.dirSupInacc | null |
dirSupClosed_Iic (a : α) : DirSupClosed (Iic a) := fun _d _ _ _a ha ↦ (isLUB_le_iff ha).2 | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | dirSupClosed_Iic | null |
dirSupInacc_iff_forall_sSup :
DirSupInacc s ↔ ∀ ⦃d⦄, d.Nonempty → DirectedOn (· ≤ ·) d → sSup d ∈ s → (d ∩ s).Nonempty := by
simp [DirSupInacc, isLUB_iff_sSup_eq] | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | dirSupInacc_iff_forall_sSup | null |
dirSupClosed_iff_forall_sSup :
DirSupClosed s ↔ ∀ ⦃d⦄, d.Nonempty → DirectedOn (· ≤ ·) d → d ⊆ s → sSup d ∈ s := by
simp [DirSupClosed, isLUB_iff_sSup_eq] | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | dirSupClosed_iff_forall_sSup | null |
scottHausdorff (α : Type*) (D : Set (Set α)) [Preorder α] : TopologicalSpace α where
IsOpen u := ∀ ⦃d : Set α⦄, d ∈ D → d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a : α⦄, IsLUB d a →
a ∈ u → ∃ b ∈ d, Ici b ∩ d ⊆ u
isOpen_univ := fun d _ ⟨b, hb⟩ _ _ _ _ ↦ ⟨b, hb, (Ici b ∩ d).subset_univ⟩
isOpen_inter s t hs ht d h... | def | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | scottHausdorff | The Scott-Hausdorff topology.
A set `u` is open in the Scott-Hausdorff topology iff when the least upper bound of a directed set
`d` lies in `u` then there is a tail of `d` which is a subset of `u`. |
IsScottHausdorff : Prop where
topology_eq_scottHausdorff : ‹TopologicalSpace α› = scottHausdorff α D | class | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | IsScottHausdorff | Predicate for an ordered topological space to be equipped with its Scott-Hausdorff topology.
A set `u` is open in the Scott-Hausdorff topology iff when the least upper bound of a directed set
`d` lies in `u` then there is a tail of `d` which is a subset of `u`. |
topology_eq [IsScottHausdorff α D] : ‹_› = scottHausdorff α D := topology_eq_scottHausdorff
variable {α D} | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | topology_eq | null |
isOpen_iff [IsScottHausdorff α D] :
IsOpen s ↔ ∀ ⦃d : Set α⦄, d ∈ D → d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a : α⦄, IsLUB d a →
a ∈ s → ∃ b ∈ d, Ici b ∩ d ⊆ s := by
simp [topology_eq_scottHausdorff (α := α) (D := D), IsOpen, scottHausdorff] | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | isOpen_iff | null |
dirSupInaccOn_of_isOpen [IsScottHausdorff α D] (h : IsOpen s) : DirSupInaccOn D s :=
fun d hd₀ hd₁ hd₂ a hda hd₃ ↦ by
obtain ⟨b, hbd, hb⟩ := isOpen_iff.mp h hd₀ hd₁ hd₂ hda hd₃; exact ⟨b, hbd, hb ⟨le_rfl, hbd⟩⟩ | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | dirSupInaccOn_of_isOpen | null |
dirSupClosed_of_isClosed [IsScottHausdorff α univ] (h : IsClosed s) : DirSupClosed s := by
apply DirSupInacc.of_compl
rw [← dirSupInaccOn_univ]
exact (dirSupInaccOn_of_isOpen h.isOpen_compl) | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | dirSupClosed_of_isClosed | null |
isOpen_of_isLowerSet (h : IsLowerSet s) : IsOpen s :=
(isOpen_iff (D := univ)).2 fun _d _ ⟨b, hb⟩ _ _ hda ha ↦
⟨b, hb, fun _ hc ↦ h (mem_upperBounds.1 hda.1 _ hc.2) ha⟩ | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | isOpen_of_isLowerSet | null |
isClosed_of_isUpperSet (h : IsUpperSet s) : IsClosed s :=
isOpen_compl_iff.1 <| isOpen_of_isLowerSet h.compl | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | isClosed_of_isUpperSet | null |
scott (α : Type*) (D : Set (Set α)) [Preorder α] : TopologicalSpace α :=
upperSet α ⊔ scottHausdorff α D | def | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | scott | The Scott topology.
It is defined as the join of the topology of upper sets and the Scott-Hausdorff topology. |
upperSet_le_scott [Preorder α] : upperSet α ≤ scott α univ := le_sup_left | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | upperSet_le_scott | null |
scottHausdorff_le_scott [Preorder α] : scottHausdorff α univ ≤ scott α univ := le_sup_right
variable (α) (D) [Preorder α] [TopologicalSpace α] | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | scottHausdorff_le_scott | null |
IsScott : Prop where
topology_eq_scott : ‹TopologicalSpace α› = scott α D | class | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | IsScott | Predicate for an ordered topological space to be equipped with its Scott topology.
The Scott topology is defined as the join of the topology of upper sets and the Scott Hausdorff
topology. |
topology_eq [IsScott α D] : ‹_› = scott α D := topology_eq_scott
variable {α} {D} {s : Set α} {a : α} | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | topology_eq | null |
isOpen_iff_isUpperSet_and_scottHausdorff_open [IsScott α D] :
IsOpen s ↔ IsUpperSet s ∧ IsOpen[scottHausdorff α D] s := by rw [topology_eq α D]; rfl | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | isOpen_iff_isUpperSet_and_scottHausdorff_open | null |
isOpen_iff_isUpperSet_and_dirSupInaccOn [IsScott α D] :
IsOpen s ↔ IsUpperSet s ∧ DirSupInaccOn D s := by
rw [isOpen_iff_isUpperSet_and_scottHausdorff_open (D := D)]
refine and_congr_right fun h ↦
⟨@IsScottHausdorff.dirSupInaccOn_of_isOpen _ _ _ (scottHausdorff α D) _ _,
fun h' d d₀ d₁ d₂ _ d₃ ha ↦ ?_... | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | isOpen_iff_isUpperSet_and_dirSupInaccOn | null |
isClosed_iff_isLowerSet_and_dirSupClosed [IsScott α univ] :
IsClosed s ↔ IsLowerSet s ∧ DirSupClosed s := by
rw [← isOpen_compl_iff, isOpen_iff_isUpperSet_and_dirSupInaccOn (D := univ), isUpperSet_compl,
dirSupInaccOn_univ, dirSupInacc_compl] | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | isClosed_iff_isLowerSet_and_dirSupClosed | null |
isUpperSet_of_isOpen [IsScott α D] : IsOpen s → IsUpperSet s := fun h ↦
(isOpen_iff_isUpperSet_and_scottHausdorff_open (D := D).mp h).left | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | isUpperSet_of_isOpen | null |
isLowerSet_of_isClosed [IsScott α univ] : IsClosed s → IsLowerSet s := fun h ↦
(isClosed_iff_isLowerSet_and_dirSupClosed.mp h).left | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | isLowerSet_of_isClosed | null |
dirSupClosed_of_isClosed [IsScott α univ] : IsClosed s → DirSupClosed s := fun h ↦
(isClosed_iff_isLowerSet_and_dirSupClosed.mp h).right | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | dirSupClosed_of_isClosed | null |
lowerClosure_subset_closure [IsScott α univ] : ↑(lowerClosure s) ⊆ closure s := by
convert closure.mono (@upperSet_le_scott α _)
· rw [@IsUpperSet.closure_eq_lowerClosure α _ (upperSet α) ?_ s]
infer_instance
· exact topology_eq α univ | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | lowerClosure_subset_closure | null |
@[simp] closure_singleton [IsScott α univ] : closure {a} = Iic a := le_antisymm
(closure_minimal (by rw [singleton_subset_iff, mem_Iic]) isClosed_Iic) <| by
rw [← LowerSet.coe_Iic, ← lowerClosure_singleton]
apply lowerClosure_subset_closure
variable [Preorder β] [TopologicalSpace β] [IsScott β univ] {f : α → ... | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | closure_singleton | The closure of a singleton `{a}` in the Scott topology is the right-closed left-infinite interval
`(-∞,a]`. |
monotone_of_continuous [IsScott α D] (hf : Continuous f) : Monotone f := fun _ b hab ↦ by
by_contra h
simpa only [mem_compl_iff, mem_preimage, mem_Iic, le_refl, not_true]
using isUpperSet_of_isOpen (D := D) ((isOpen_compl_iff.2 isClosed_Iic).preimage hf) hab h
@[simp] lemma scottContinuousOn_iff_continuous {D :... | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | monotone_of_continuous | null |
WithScott (α : Type*) := α | def | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | WithScott | The Scott topology on a partial order is T₀.
-/
-- see Note [lower instance priority]
instance (priority := 90) : T0Space α :=
(t0Space_iff_inseparable α).2 fun x y h ↦ Iic_injective <| by
simpa only [inseparable_iff_closure_eq, IsScott.closure_singleton] using h
end PartialOrder
section CompleteLinearOrder
va... |
@[match_pattern] toScott : α ≃ WithScott α := Equiv.refl _ | def | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | toScott | `toScott` is the identity function to the `WithScott` of a type. |
@[match_pattern] ofScott : WithScott α ≃ α := Equiv.refl _
@[simp] lemma toScott_symm_eq : (@toScott α).symm = ofScott := rfl
@[simp] lemma ofScott_symm_eq : (@ofScott α).symm = toScott := rfl
@[simp] lemma toScott_ofScott (a : WithScott α) : toScott (ofScott a) = a := rfl
@[simp] lemma ofScott_toScott (a : α) : ofScot... | def | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | ofScott | `ofScott` is the identity function from the `WithScott` of a type. |
toScott_inj {a b : α} : toScott a = toScott b ↔ a = b := Iff.rfl | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | toScott_inj | null |
ofScott_inj {a b : WithScott α} : ofScott a = ofScott b ↔ a = b := Iff.rfl | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | ofScott_inj | null |
@[elab_as_elim, cases_eliminator, induction_eliminator]
protected rec {β : WithScott α → Sort _}
(h : ∀ a, β (toScott a)) : ∀ a, β a := fun a ↦ h (ofScott a) | def | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | rec | A recursor for `WithScott`. Use as `induction x`. |
isOpen_iff_isUpperSet_and_scottHausdorff_open' {u : Set α} :
IsOpen (WithScott.ofScott ⁻¹' u) ↔ IsUpperSet u ∧ (scottHausdorff α univ).IsOpen u := Iff.rfl | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | isOpen_iff_isUpperSet_and_scottHausdorff_open' | null |
scottHausdorff_le_lower : scottHausdorff α univ ≤ lower α :=
fun s h => IsScottHausdorff.isOpen_of_isLowerSet (t := scottHausdorff α univ)
<| (@IsLower.isLowerSet_of_isOpen (Topology.WithLower α) _ _ _ s h)
variable [TopologicalSpace α] | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | scottHausdorff_le_lower | null |
IsScott.withScottHomeomorph [IsScott α univ] : WithScott α ≃ₜ α :=
WithScott.ofScott.toHomeomorphOfIsInducing ⟨IsScott.topology_eq α univ ▸ induced_id.symm⟩ | def | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | IsScott.withScottHomeomorph | If `α` is equipped with the Scott topology, then it is homeomorphic to `WithScott α`. |
IsScott.scottHausdorff_le [IsScott α univ] :
scottHausdorff α univ ≤ ‹TopologicalSpace α› := by
rw [IsScott.topology_eq α univ, scott]; exact le_sup_right | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | IsScott.scottHausdorff_le | null |
IsLower.scottHausdorff_le [IsLower α] : scottHausdorff α univ ≤ ‹TopologicalSpace α› :=
fun _ h ↦
IsScottHausdorff.isOpen_of_isLowerSet (t := scottHausdorff α univ)
<| IsLower.isLowerSet_of_isOpen h | lemma | Topology | [
"Mathlib.Order.ScottContinuity",
"Mathlib.Topology.Order.UpperLowerSetTopology"
] | Mathlib/Topology/Order/ScottTopology.lean | IsLower.scottHausdorff_le | null |
@[simp]
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 | Topology | [
"Mathlib.Order.Interval.Set.OrdConnectedComponent",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Order/T5.lean | ordConnectedComponent_mem_nhds | null |
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... | theorem | Topology | [
"Mathlib.Order.Interval.Set.OrdConnectedComponent",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Order/T5.lean | compl_ordConnectedSection_ordSeparatingSet_mem_nhdsGE | null |
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_... | theorem | Topology | [
"Mathlib.Order.Interval.Set.OrdConnectedComponent",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Order/T5.lean | compl_ordConnectedSection_ordSeparatingSet_mem_nhdsLE | null |
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_nh... | theorem | Topology | [
"Mathlib.Order.Interval.Set.OrdConnectedComponent",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Order/T5.lean | compl_ordConnectedSection_ordSeparatingSet_mem_nhds | null |
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 | Topology | [
"Mathlib.Order.Interval.Set.OrdConnectedComponent",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Order/T5.lean | ordT5Nhd_mem_nhdsSet | null |
upperSet (α : Type*) [Preorder α] : TopologicalSpace α where
IsOpen := IsUpperSet
isOpen_univ := isUpperSet_univ
isOpen_inter _ _ := IsUpperSet.inter
isOpen_sUnion _ := isUpperSet_sUnion | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | upperSet | Topology whose open sets are upper sets.
Note: In general the upper set topology does not coincide with the upper topology. |
lowerSet (α : Type*) [Preorder α] : TopologicalSpace α where
IsOpen := IsLowerSet
isOpen_univ := isLowerSet_univ
isOpen_inter _ _ := IsLowerSet.inter
isOpen_sUnion _ := isLowerSet_sUnion | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | lowerSet | Topology whose open sets are lower sets.
Note: In general the lower set topology does not coincide with the lower topology. |
WithUpperSet (α : Type*) := α | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | WithUpperSet | Type synonym for a preorder equipped with the upper set topology. |
@[match_pattern] toUpperSet : α ≃ WithUpperSet α := Equiv.refl _ | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | toUpperSet | `toUpperSet` is the identity function to the `WithUpperSet` of a type. |
@[match_pattern] ofUpperSet : WithUpperSet α ≃ α := Equiv.refl _
@[simp] lemma toUpperSet_symm : (@toUpperSet α).symm = ofUpperSet := rfl
@[simp] lemma ofUpperSet_symm : (@ofUpperSet α).symm = toUpperSet := rfl
@[simp] lemma toUpperSet_ofUpperSet (a : WithUpperSet α) : toUpperSet (ofUpperSet a) = a := rfl
@[simp] lemma... | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | ofUpperSet | `ofUpperSet` is the identity function from the `WithUpperSet` of a type. |
toUpperSet_inj {a b : α} : toUpperSet a = toUpperSet b ↔ a = b := Iff.rfl | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | toUpperSet_inj | null |
ofUpperSet_inj {a b : WithUpperSet α} : ofUpperSet a = ofUpperSet b ↔ a = b := Iff.rfl | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | ofUpperSet_inj | null |
@[elab_as_elim, cases_eliminator, induction_eliminator]
protected rec {β : WithUpperSet α → Sort*} (h : ∀ a, β (toUpperSet a)) : ∀ a, β a :=
fun a => h (ofUpperSet a) | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | rec | A recursor for `WithUpperSet`. Use as `induction x`. |
ofUpperSet_le_iff {a b : WithUpperSet α} : ofUpperSet a ≤ ofUpperSet b ↔ a ≤ b := Iff.rfl | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | ofUpperSet_le_iff | null |
toUpperSet_le_iff {a b : α} : toUpperSet a ≤ toUpperSet b ↔ a ≤ b := Iff.rfl | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | toUpperSet_le_iff | null |
ofUpperSetOrderIso : WithUpperSet α ≃o α where
toEquiv := ofUpperSet
map_rel_iff' := ofUpperSet_le_iff | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | ofUpperSetOrderIso | `ofUpperSet` as an `OrderIso` |
toUpperSetOrderIso : α ≃o WithUpperSet α where
toEquiv := toUpperSet
map_rel_iff' := toUpperSet_le_iff | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | toUpperSetOrderIso | `toUpperSet` as an `OrderIso` |
WithLowerSet (α : Type*) := α | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | WithLowerSet | Type synonym for a preorder equipped with the lower set topology. |
@[match_pattern] toLowerSet : α ≃ WithLowerSet α := Equiv.refl _ | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | toLowerSet | `toLowerSet` is the identity function to the `WithLowerSet` of a type. |
@[match_pattern] ofLowerSet : WithLowerSet α ≃ α := Equiv.refl _
@[simp] lemma toLowerSet_symm : (@toLowerSet α).symm = ofLowerSet := rfl
@[simp] lemma ofLowerSet_symm : (@ofLowerSet α).symm = toLowerSet := rfl
@[simp] lemma toLowerSet_ofLowerSet (a : WithLowerSet α) : toLowerSet (ofLowerSet a) = a := rfl
@[simp] lemma... | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | ofLowerSet | `ofLowerSet` is the identity function from the `WithLowerSet` of a type. |
toLowerSet_inj {a b : α} : toLowerSet a = toLowerSet b ↔ a = b := Iff.rfl | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | toLowerSet_inj | null |
ofLowerSet_inj {a b : WithLowerSet α} : ofLowerSet a = ofLowerSet b ↔ a = b := Iff.rfl | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | ofLowerSet_inj | null |
@[elab_as_elim, cases_eliminator, induction_eliminator]
protected rec {β : WithLowerSet α → Sort*} (h : ∀ a, β (toLowerSet a)) : ∀ a, β a :=
fun a => h (ofLowerSet a) | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | rec | A recursor for `WithLowerSet`. Use as `induction x`. |
ofLowerSet_le_iff {a b : WithLowerSet α} : ofLowerSet a ≤ ofLowerSet b ↔ a ≤ b := Iff.rfl | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | ofLowerSet_le_iff | null |
toLowerSet_le_iff {a b : α} : toLowerSet a ≤ toLowerSet b ↔ a ≤ b := Iff.rfl | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | toLowerSet_le_iff | null |
ofLowerSetOrderIso : WithLowerSet α ≃o α where
toEquiv := ofLowerSet
map_rel_iff' := ofLowerSet_le_iff | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | ofLowerSetOrderIso | `ofLowerSet` as an `OrderIso` |
toLowerSetOrderIso : α ≃o WithLowerSet α where
toEquiv := toLowerSet
map_rel_iff' := toLowerSet_le_iff | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | toLowerSetOrderIso | `toLowerSet` as an `OrderIso` |
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 | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | WithUpperSet.toDualHomeomorph | The Upper Set topology is homeomorphic to the Lower Set topology on the dual order |
protected IsUpperSet (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
topology_eq_upperSetTopology : t = upperSet α
attribute [nolint docBlame] IsUpperSet.topology_eq_upperSetTopology | class | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | IsUpperSet | 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. |
protected IsLowerSet (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
topology_eq_lowerSetTopology : t = lowerSet α
attribute [nolint docBlame] IsLowerSet.topology_eq_lowerSetTopology | class | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | IsLowerSet | 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. |
topology_eq : ‹_› = upperSet α := topology_eq_upperSetTopology
variable {α} | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | topology_eq | null |
_root_.OrderDual.instIsLowerSet [Preorder α] [TopologicalSpace α] [Topology.IsUpperSet α] :
Topology.IsLowerSet αᵒᵈ where
topology_eq_lowerSetTopology := by ext; rw [IsUpperSet.topology_eq α] | instance | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | _root_.OrderDual.instIsLowerSet | null |
WithUpperSetHomeomorph : WithUpperSet α ≃ₜ α :=
WithUpperSet.ofUpperSet.toHomeomorphOfIsInducing ⟨topology_eq α ▸ induced_id.symm⟩ | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | WithUpperSetHomeomorph | If `α` is equipped with the upper set topology, then it is homeomorphic to
`WithUpperSet α`. |
isOpen_iff_isUpperSet : IsOpen s ↔ IsUpperSet s := by
rw [topology_eq α]
rfl | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | isOpen_iff_isUpperSet | null |
toAlexandrovDiscrete : AlexandrovDiscrete α where
isOpen_sInter S := by simpa only [isOpen_iff_isUpperSet] using isUpperSet_sInter (α := α) | instance | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | toAlexandrovDiscrete | null |
isClosed_iff_isLower : IsClosed s ↔ IsLowerSet s := by
rw [← isOpen_compl_iff, isOpen_iff_isUpperSet,
isLowerSet_compl.symm, compl_compl] | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | isClosed_iff_isLower | null |
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 | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | closure_eq_lowerClosure | null |
@[simp] closure_singleton {a : α} : closure {a} = Iic a := by
rw [closure_eq_lowerClosure, lowerClosure_singleton]
rfl | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | closure_singleton | The closure of a singleton `{a}` in the upper set topology is the right-closed left-infinite
interval (-∞,a]. |
specializes_iff_le {a b : α} : a ⤳ b ↔ b ≤ a := by
simp only [specializes_iff_closure_subset, closure_singleton, Iic_subset_Iic] | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | specializes_iff_le | null |
nhdsKer_eq_upperClosure (s : Set α) : nhdsKer s = ↑(upperClosure s) := by
ext; simp [mem_nhdsKer_iff_specializes, specializes_iff_le]
@[simp] lemma nhdsKer_singleton (a : α) : nhdsKer {a} = Ici a := by
rw [nhdsKer_eq_upperClosure, upperClosure_singleton, UpperSet.coe_Ici] | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | nhdsKer_eq_upperClosure | null |
nhds_eq_principal_Ici (a : α) : 𝓝 a = 𝓟 (Ici a) := by
rw [← principal_nhdsKer_singleton, nhdsKer_singleton] | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | nhds_eq_principal_Ici | null |
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