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 ⌀ |
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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_lower_of_not_le h
exact ⟨U, hU, Or.inr hU', hx, hy⟩ | 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] [TopologicalSpace β] [TopologicalSpace γ]
@[continuity, fun_prop] | 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 <| continuous_projIcc.continuousAt.comp hg | 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 ⟨C, Cmem, Cge⟩ : ∃ C ∈ Icc a b, ∀ x ∈ Icc a b, f x ≤ f C :=
isCompact_Icc.exists_isMaxOn ne hfc
by_cases hc : f c = f a
· by_cases hC : f C = f a
· have : ∀ x ∈ Icc a b, f x = f a := fun x hx => le_antisymm (hC ▸ Cge x hx) (hc ▸ cle x hx)
rcases nonempty_Ioo.2 hab with ⟨c', hc'⟩
refine ⟨c', hc', Or.inl fun x hx ↦ ?_⟩
simp only [mem_setOf_eq, this x hx, this c' (Ioo_subset_Icc_self hc'), le_rfl]
· refine ⟨C, ⟨lt_of_le_of_ne Cmem.1 <| mt ?_ hC, lt_of_le_of_ne Cmem.2 <| mt ?_ hC⟩, Or.inr Cge⟩
exacts [fun h => by rw [h], fun h => by rw [h, hfI]]
· refine ⟨c, ⟨lt_of_le_of_ne cmem.1 <| mt ?_ hc, lt_of_le_of_ne cmem.2 <| mt ?_ hc⟩, Or.inl cle⟩
exacts [fun h => by rw [h], fun h => by rw [h, hfI]] | 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, IsExtrOn (extendFrom (Ioo a b) f) (Icc a b) c :=
exists_Ioo_extr_on_Icc hab (continuousOn_Icc_extendFrom_Ioo hab.ne hfc ha hb)
((eq_lim_at_left_extendFrom_Ioo hab ha).trans (eq_lim_at_right_extendFrom_Ioo hab hb).symm)
exact ⟨c, hc, (hfc.on_subset Ioo_subset_Icc_self).congr h (h hc)⟩ | 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 function between `OmegaCompletePartialOrder`s is always
`OmegaCompletePartialOrder.Continuous'` (`OmegaCompletePartialOrder.ScottContinuous.continuous'`).
The converse is true in some special cases, but not in general
([Domain Theory, 2.2.4][abramsky_gabbay_maibaum_1994]). | 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 → DirSupInaccOn D s := fun h _ _ d₂ d₃ _ hda => h d₂ d₃ hda | 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 dirSupClosed_compl : DirSupClosed sᶜ ↔ DirSupInacc s := by
rw [← dirSupInacc_compl, compl_compl]
alias ⟨DirSupInacc.of_compl, DirSupClosed.compl⟩ := dirSupInacc_compl
alias ⟨DirSupClosed.of_compl, DirSupInacc.compl⟩ := dirSupClosed_compl | 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 hd₀ hd₁ hd₂ a hd₃ ha := by
obtain ⟨b₁, hb₁d, hb₁ds⟩ := hs hd₀ hd₁ hd₂ hd₃ ha.1
obtain ⟨b₂, hb₂d, hb₂dt⟩ := ht hd₀ hd₁ hd₂ hd₃ ha.2
obtain ⟨c, hcd, hc⟩ := hd₂ b₁ hb₁d b₂ hb₂d
exact ⟨c, hcd, fun e ⟨hce, hed⟩ ↦ ⟨hb₁ds ⟨hc.1.trans hce, hed⟩, hb₂dt ⟨hc.2.trans hce, hed⟩⟩⟩
isOpen_sUnion := fun s h d hd₀ hd₁ hd₂ a hd₃ ⟨s₀, hs₀s, has₀⟩ ↦ by
obtain ⟨b, hbd, hbds₀⟩ := h s₀ hs₀s hd₀ hd₁ hd₂ hd₃ has₀
exact ⟨b, hbd, Set.subset_sUnion_of_subset s s₀ hbds₀ hs₀s⟩
variable (α) (D : Set (Set α)) [Preorder α] [TopologicalSpace α] | 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 ↦ ?_⟩
obtain ⟨b, hbd, hbu⟩ := h' d₀ d₁ d₂ d₃ ha
exact ⟨b, hbd, Subset.trans inter_subset_left (h.Ici_subset hbu)⟩ | 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 → β} | 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 : Set (Set α)} [Topology.IsScott α D]
(hD : ∀ a b : α, a ≤ b → {a, b} ∈ D) : ScottContinuousOn D f ↔ Continuous f := by
refine ⟨fun h ↦ continuous_def.2 fun u hu ↦ ?_, ?_⟩
· rw [isOpen_iff_isUpperSet_and_dirSupInaccOn (D := D)]
exact ⟨(isUpperSet_of_isOpen (D := univ) hu).preimage (h.monotone D hD),
fun t h₀ hd₁ hd₂ a hd₃ ha ↦ image_inter_nonempty_iff.mp <|
(isOpen_iff_isUpperSet_and_dirSupInaccOn (D := univ).mp hu).2 trivial (Nonempty.image f hd₁)
(directedOn_image.mpr (hd₂.mono @(h.monotone D hD))) (h h₀ hd₁ hd₂ hd₃) ha⟩
· refine fun hf t h₀ d₁ d₂ a d₃ ↦
⟨(monotone_of_continuous (D := D) hf).mem_upperBounds_image d₃.1,
fun b hb ↦ ?_⟩
by_contra h
let u := (Iic b)ᶜ
have hu : IsOpen (f ⁻¹' u) := isClosed_Iic.isOpen_compl.preimage hf
rw [isOpen_iff_isUpperSet_and_dirSupInaccOn (D := D)] at hu
obtain ⟨c, hcd, hfcb⟩ := hu.2 h₀ d₁ d₂ d₃ h
simp only [upperBounds, mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂,
mem_setOf] at hb
exact hfcb <| hb _ hcd
@[deprecated (since := "2025-07-02")]
alias scottContinuous_iff_continuous := scottContinuousOn_iff_continuous | 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
variable [CompleteLinearOrder α]
lemma isOpen_iff_Iic_compl_or_univ [TopologicalSpace α] [Topology.IsScott α univ] (U : Set α) :
IsOpen U ↔ U = univ ∨ ∃ a, (Iic a)ᶜ = U := by
constructor
· intro hU
rcases eq_empty_or_nonempty Uᶜ with eUc | neUc
· exact Or.inl (compl_empty_iff.mp eUc)
· apply Or.inr
use sSup Uᶜ
rw [compl_eq_comm, le_antisymm_iff]
exact ⟨fun _ ha ↦ le_sSup ha, (isLowerSet_of_isClosed hU.isClosed_compl).Iic_subset
(dirSupClosed_iff_forall_sSup.mp (dirSupClosed_of_isClosed hU.isClosed_compl)
neUc (isChain_of_trichotomous Uᶜ).directedOn le_rfl)⟩
· rintro (rfl | ⟨a, rfl⟩)
· exact isOpen_univ
· exact isClosed_Iic.isOpen_compl
-- N.B. A number of conditions equivalent to `scott α = upper α` are given in Gierz _et al_,
-- Chapter III, Exercise 3.23.
lemma scott_eq_upper_of_completeLinearOrder : scott α univ = upper α := by
letI := upper α
ext U
rw [@Topology.IsUpper.isTopologicalSpace_basis _ _ (upper α)
({ topology_eq_upperTopology := rfl }) U]
letI := scott α univ
rw [@isOpen_iff_Iic_compl_or_univ _ _ (scott α univ) ({ topology_eq_scott := rfl }) U]
/- The upper topology on a complete linear order is the Scott topology -/
instance [TopologicalSpace α] [IsUpper α] : IsScott α univ where
topology_eq_scott := by
rw [scott_eq_upper_of_completeLinearOrder]
exact IsUpper.topology_eq α
end CompleteLinearOrder
lemma isOpen_iff_scottContinuous_mem [Preorder α] {s : Set α} [TopologicalSpace α]
[IsScott α univ] : IsOpen s ↔ ScottContinuous fun x ↦ x ∈ s := by
rw [← scottContinuousOn_univ, scottContinuousOn_iff_continuous (fun _ _ _ ↦ by trivial)]
exact isOpen_iff_continuous_mem
end IsScott
/--
Type synonym for a preorder equipped with the Scott topology |
@[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) : ofScott (toScott a) = a := rfl | 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 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)
· exact mem_of_superset hmem' (disjoint_left.1 H)
· simp only [Set.disjoint_left, not_forall, Classical.not_not] at H
rcases H with ⟨c, ⟨hac, hcb⟩, hc⟩
have hsub' : Icc a b ⊆ ordConnectedComponent tᶜ a :=
subset_ordConnectedComponent (left_mem_Icc.2 hab) hsub
have hd : Disjoint s (ordConnectedSection (ordSeparatingSet s t)) :=
disjoint_left_ordSeparatingSet.mono_right ordConnectedSection_subset
replace hac : a < c := hac.lt_of_ne <| Ne.symm <| ne_of_mem_of_not_mem hc <|
disjoint_left.1 hd ha
filter_upwards [Ico_mem_nhdsGE hac] with x hx hx'
refine hx.2.ne (eq_of_mem_ordConnectedSection_of_uIcc_subset hx' hc ?_)
refine subset_inter (subset_iUnion₂_of_subset a ha ?_) ?_
· exact OrdConnected.uIcc_subset inferInstance (hsub' ⟨hx.1, hx.2.le.trans hcb⟩)
(hsub' ⟨hac.le, hcb⟩)
· rcases mem_iUnion₂.1 (ordConnectedSection_subset hx').2 with ⟨y, hyt, hxy⟩
refine subset_iUnion₂_of_subset y hyt (OrdConnected.uIcc_subset inferInstance hxy ?_)
refine subset_ordConnectedComponent left_mem_uIcc hxy ?_
suffices c < y by
rw [uIcc_of_ge (hx.2.trans this).le]
exact ⟨hx.2.le, this.le⟩
refine lt_of_not_ge fun hyc => ?_
have hya : y < a := not_le.1 fun hay => hsub ⟨hay, hyc.trans hcb⟩ hyt
exact hxy (Icc_subset_uIcc ⟨hya.le, hx.1⟩) ha | 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_ordConnectedSection] using
compl_ordConnectedSection_ordSeparatingSet_mem_nhdsGE hd' ha' | 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_nhdsGE hd ha⟩ | 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 ofUpperSet_toUpperSet (a : α) : ofUpperSet (toUpperSet a) = a := rfl | 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 ofLowerSet_toLowerSet (a : α) : ofLowerSet (toLowerSet a) = a := rfl | 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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