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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BaireMeasurableSet.iff_residualEq_isOpen :
BaireMeasurableSet s ↔ ∃ u : Set α, (IsOpen u) ∧ s =ᵇ u :=
⟨fun h => h.residualEq_isOpen, fun ⟨_, uo, ueq⟩ => uo.baireMeasurableSet.congr ueq.symm⟩ | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | BaireMeasurableSet.iff_residualEq_isOpen | A set is Baire measurable if and only if it differs from some open set by a meager set. |
tendsto_residual_of_isOpenMap (hc : Continuous f) (ho : IsOpenMap f) :
Tendsto f (residual α) (residual β) := by
apply le_countableGenerate_iff_of_countableInterFilter.mpr
rintro t ⟨ht, htd⟩
exact residual_of_dense_open (ht.preimage hc) (htd.preimage ho) | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | tendsto_residual_of_isOpenMap | null |
IsMeagre.preimage_of_isOpenMap (hc : Continuous f) (ho : IsOpenMap f)
{s : Set β} (h : IsMeagre s) : IsMeagre (f ⁻¹' s) :=
tendsto_residual_of_isOpenMap hc ho h | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | IsMeagre.preimage_of_isOpenMap | The preimage of a meager set under a continuous open map is meager. |
BaireMeasurableSet.preimage (hc : Continuous f) (ho : IsOpenMap f)
{s : Set β} (h : BaireMeasurableSet s) : BaireMeasurableSet (f ⁻¹' s) := by
rcases h with ⟨u, hu, hsu⟩
refine ⟨f ⁻¹' u, ?_, hsu.filter_mono <| tendsto_residual_of_isOpenMap hc ho⟩
borelize α β
exact hc.measurable hu | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | BaireMeasurableSet.preimage | The preimage of a `BaireMeasurableSet` under a continuous open map is Baire measurable. |
Homeomorph.residual_map_eq (h : α ≃ₜ β) : (residual α).map h = residual β := by
refine le_antisymm (tendsto_residual_of_isOpenMap h.continuous h.isOpenMap) (le_map ?_)
simp_rw [← preimage_symm]
exact tendsto_residual_of_isOpenMap h.symm.continuous h.symm.isOpenMap | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | Homeomorph.residual_map_eq | null |
dense_iInter_of_isOpen_nat {f : ℕ → Set X} (ho : ∀ n, IsOpen (f n))
(hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) :=
BaireSpace.baire_property f ho hd | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | dense_iInter_of_isOpen_nat | Definition of a Baire space. |
dense_sInter_of_isOpen {S : Set (Set X)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) := by
rcases S.eq_empty_or_nonempty with h | h
· simp [h]
· rcases hS.exists_eq_range h with ⟨f, rfl⟩
exact dense_iInter_of_isOpen_nat (forall_mem_range.1 ho) (forall_mem_range.1 hd) | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | dense_sInter_of_isOpen | Baire theorem: a countable intersection of dense open sets is dense. Formulated here with ⋂₀. |
dense_biInter_of_isOpen {S : Set α} {f : α → Set X} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) := by
rw [← sInter_image]
refine dense_sInter_of_isOpen ?_ (hS.image _) ?_ <;> rwa [forall_mem_image] | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | dense_biInter_of_isOpen | Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable set in any type. |
dense_iInter_of_isOpen [Countable ι] {f : ι → Set X} (ho : ∀ i, IsOpen (f i))
(hd : ∀ i, Dense (f i)) : Dense (⋂ s, f s) :=
dense_sInter_of_isOpen (forall_mem_range.2 ho) (countable_range _) (forall_mem_range.2 hd) | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | dense_iInter_of_isOpen | Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable type. |
mem_residual {s : Set X} : s ∈ residual X ↔ ∃ t ⊆ s, IsGδ t ∧ Dense t := by
constructor
· rw [mem_residual_iff]
rintro ⟨S, hSo, hSd, Sct, Ss⟩
refine ⟨_, Ss, ⟨_, fun t ht => hSo _ ht, Sct, rfl⟩, ?_⟩
exact dense_sInter_of_isOpen hSo Sct hSd
rintro ⟨t, ts, ho, hd⟩
exact mem_of_superset (residual_of_dense_Gδ ho hd) ts | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | mem_residual | A set is residual (comeagre) if and only if it includes a dense `Gδ` set. |
eventually_residual {p : X → Prop} :
(∀ᶠ x in residual X, p x) ↔ ∃ t : Set X, IsGδ t ∧ Dense t ∧ ∀ x ∈ t, p x := by
simp only [Filter.Eventually, mem_residual, subset_def, mem_setOf_eq]
tauto | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | eventually_residual | A property holds on a residual (comeagre) set if and only if it holds on some dense `Gδ` set. |
dense_of_mem_residual {s : Set X} (hs : s ∈ residual X) : Dense s :=
let ⟨_, hts, _, hd⟩ := mem_residual.1 hs
hd.mono hts | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | dense_of_mem_residual | null |
not_isMeagre_of_isOpen {s : Set X} (hs : IsOpen s) (hne : s.Nonempty) : ¬ IsMeagre s := by
intro h
obtain ⟨x, hx, hxc⟩ :=
(dense_of_mem_residual (by rwa [IsMeagre] at h)).inter_open_nonempty s hs hne
exact hxc hx | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | not_isMeagre_of_isOpen | In a Baire space, every nonempty open set is non‐meagre,
that is, it cannot be written as a countable union of nowhere‐dense sets. |
dense_sInter_of_Gδ {S : Set (Set X)} (ho : ∀ s ∈ S, IsGδ s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) :=
dense_of_mem_residual ((countable_sInter_mem hS).mpr
(fun _ hs => residual_of_dense_Gδ (ho _ hs) (hd _ hs))) | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | dense_sInter_of_Gδ | Baire theorem: a countable intersection of dense Gδ sets is dense. Formulated here with ⋂₀. |
dense_iInter_of_Gδ [Countable ι] {f : ι → Set X} (ho : ∀ s, IsGδ (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) :=
dense_sInter_of_Gδ (forall_mem_range.2 ‹_›) (countable_range _) (forall_mem_range.2 ‹_›) | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | dense_iInter_of_Gδ | Baire theorem: a countable intersection of dense Gδ sets is dense. Formulated here with
an index set which is a countable type. |
dense_biInter_of_Gδ {S : Set α} {f : ∀ x ∈ S, Set X} (ho : ∀ s (H : s ∈ S), IsGδ (f s H))
(hS : S.Countable) (hd : ∀ s (H : s ∈ S), Dense (f s H)) : Dense (⋂ s ∈ S, f s ‹_›) := by
rw [biInter_eq_iInter]
haveI := hS.to_subtype
exact dense_iInter_of_Gδ (fun s => ho s s.2) fun s => hd s s.2 | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | dense_biInter_of_Gδ | Baire theorem: a countable intersection of dense Gδ sets is dense. Formulated here with
an index set which is a countable set in any type. |
Dense.inter_of_Gδ {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) (hsc : Dense s)
(htc : Dense t) : Dense (s ∩ t) := by
rw [inter_eq_iInter]
apply dense_iInter_of_Gδ <;> simp [Bool.forall_bool, *] | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | Dense.inter_of_Gδ | Baire theorem: the intersection of two dense Gδ sets is dense. |
IsGδ.dense_iUnion_interior_of_closed [Countable ι] {s : Set X} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set X} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ i, interior (f i)) := by
let g i := (frontier (f i))ᶜ
have hgo : ∀ i, IsOpen (g i) := fun i => isClosed_frontier.isOpen_compl
have hgd : Dense (⋂ i, g i) := by
refine dense_iInter_of_isOpen hgo fun i x => ?_
rw [closure_compl, interior_frontier (hc _)]
exact id
refine (hd.inter_of_Gδ hs (.iInter_of_isOpen fun i => (hgo i)) hgd).mono ?_
rintro x ⟨hxs, hxg⟩
rw [mem_iInter] at hxg
rcases mem_iUnion.1 (hU hxs) with ⟨i, hi⟩
exact mem_iUnion.2 ⟨i, self_diff_frontier (f i) ▸ ⟨hi, hxg _⟩⟩ | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | IsGδ.dense_iUnion_interior_of_closed | If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. |
IsGδ.dense_biUnion_interior_of_closed {t : Set α} {s : Set X} (hs : IsGδ s) (hd : Dense s)
(ht : t.Countable) {f : α → Set X} (hc : ∀ i ∈ t, IsClosed (f i)) (hU : s ⊆ ⋃ i ∈ t, f i) :
Dense (⋃ i ∈ t, interior (f i)) := by
haveI := ht.to_subtype
simp only [biUnion_eq_iUnion, SetCoe.forall'] at *
exact hs.dense_iUnion_interior_of_closed hd hc hU | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | IsGδ.dense_biUnion_interior_of_closed | If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with a union over a countable set in any type. |
IsGδ.dense_sUnion_interior_of_closed {T : Set (Set X)} {s : Set X} (hs : IsGδ s)
(hd : Dense s) (hc : T.Countable) (hc' : ∀ t ∈ T, IsClosed t) (hU : s ⊆ ⋃₀ T) :
Dense (⋃ t ∈ T, interior t) :=
hs.dense_biUnion_interior_of_closed hd hc hc' <| by rwa [← sUnion_eq_biUnion] | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | IsGδ.dense_sUnion_interior_of_closed | If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃₀`. |
dense_biUnion_interior_of_closed {S : Set α} {f : α → Set X} (hc : ∀ s ∈ S, IsClosed (f s))
(hS : S.Countable) (hU : ⋃ s ∈ S, f s = univ) : Dense (⋃ s ∈ S, interior (f s)) :=
IsGδ.univ.dense_biUnion_interior_of_closed dense_univ hS hc hU.ge | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | dense_biUnion_interior_of_closed | Baire theorem: if countably many closed sets cover the whole space, then their interiors
are dense. Formulated here with an index set which is a countable set in any type. |
dense_sUnion_interior_of_closed {S : Set (Set X)} (hc : ∀ s ∈ S, IsClosed s)
(hS : S.Countable) (hU : ⋃₀ S = univ) : Dense (⋃ s ∈ S, interior s) :=
IsGδ.univ.dense_sUnion_interior_of_closed dense_univ hS hc hU.ge | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | dense_sUnion_interior_of_closed | Baire theorem: if countably many closed sets cover the whole space, then their interiors
are dense. Formulated here with `⋃₀`. |
dense_iUnion_interior_of_closed [Countable ι] {f : ι → Set X} (hc : ∀ i, IsClosed (f i))
(hU : ⋃ i, f i = univ) : Dense (⋃ i, interior (f i)) :=
IsGδ.univ.dense_iUnion_interior_of_closed dense_univ hc hU.ge
variable [Nonempty X] | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | dense_iUnion_interior_of_closed | Baire theorem: if countably many closed sets cover the whole space, then their interiors
are dense. Formulated here with an index set which is a countable type. |
nonempty_interior_of_iUnion_of_closed [Countable ι] {f : ι → Set X}
(hc : ∀ i, IsClosed (f i)) (hU : ⋃ i, f i = univ) : ∃ i, (interior <| f i).Nonempty := by
simpa using (dense_iUnion_interior_of_closed hc hU).nonempty | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | nonempty_interior_of_iUnion_of_closed | One of the most useful consequences of Baire theorem: if a countable union of closed sets
covers the space, then one of the sets has nonempty interior. |
not_isMeagre_of_isGδ_of_dense {s : Set X} (hs : IsGδ s) (hd : Dense s) :
¬ IsMeagre s := by
intro h
rcases (mem_residual).1 h with ⟨t, hts, htG, hd'⟩
rcases (hd.inter_of_Gδ hs htG hd').nonempty with ⟨x, hx₁, hx₂⟩
exact hts hx₂ hx₁ | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | not_isMeagre_of_isGδ_of_dense | In a nonempty Baire space, any dense `Gδ` set is not meagre. |
not_isMeagre_of_mem_residual {s : Set X} (hs : s ∈ residual X) :
¬ IsMeagre s := by
rcases (mem_residual (X := X)).1 hs with ⟨t, ht_sub, htGδ, ht_dense⟩
intro hs_meagre
exact not_isMeagre_of_isGδ_of_dense (X := X) htGδ ht_dense (hs_meagre.mono ht_sub) | theorem | Topology | [
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Baire/Lemmas.lean | not_isMeagre_of_mem_residual | In a nonempty Baire space, a residual set is not meagre. |
Absorbs (s t : Set α) : Prop :=
∀ᶠ a in cobounded M, t ⊆ a • s | def | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | Absorbs | A set `s` absorbs another set `t` if `t` is contained in all scalings of `s`
by all but a bounded set of elements. |
Absorbent (s : Set α) : Prop :=
∀ x, Absorbs M s {x} | def | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | Absorbent | A set is *absorbent* if it absorbs every singleton. |
protected empty : Absorbs M s ∅ := by simp [Absorbs] | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | empty | null |
protected eventually (h : Absorbs M s t) : ∀ᶠ a in cobounded M, t ⊆ a • s := h
@[simp] lemma of_boundedSpace [BoundedSpace M] : Absorbs M s t := by simp [Absorbs] | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | eventually | null |
mono_left (h : Absorbs M s₁ t) (hs : s₁ ⊆ s₂) : Absorbs M s₂ t :=
h.mono fun _a ha ↦ ha.trans <| smul_set_mono hs | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | mono_left | null |
mono_right (h : Absorbs M s t₁) (ht : t₂ ⊆ t₁) : Absorbs M s t₂ :=
h.mono fun _ ↦ ht.trans | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | mono_right | null |
mono (h : Absorbs M s₁ t₁) (hs : s₁ ⊆ s₂) (ht : t₂ ⊆ t₁) : Absorbs M s₂ t₂ :=
(h.mono_left hs).mono_right ht
@[simp] | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | mono | null |
_root_.absorbs_union : Absorbs M s (t₁ ∪ t₂) ↔ Absorbs M s t₁ ∧ Absorbs M s t₂ := by
simp [Absorbs] | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | _root_.absorbs_union | null |
protected union (h₁ : Absorbs M s t₁) (h₂ : Absorbs M s t₂) : Absorbs M s (t₁ ∪ t₂) :=
absorbs_union.2 ⟨h₁, h₂⟩ | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | union | null |
_root_.Set.Finite.absorbs_sUnion {T : Set (Set α)} (hT : T.Finite) :
Absorbs M s (⋃₀ T) ↔ ∀ t ∈ T, Absorbs M s t := by
simp [Absorbs, hT] | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | _root_.Set.Finite.absorbs_sUnion | null |
protected sUnion (hT : T.Finite) (hs : ∀ t ∈ T, Absorbs M s t) :
Absorbs M s (⋃₀ T) :=
hT.absorbs_sUnion.2 hs
@[simp] | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | sUnion | null |
_root_.absorbs_iUnion {ι : Sort*} [Finite ι] {t : ι → Set α} :
Absorbs M s (⋃ i, t i) ↔ ∀ i, Absorbs M s (t i) :=
(finite_range t).absorbs_sUnion.trans forall_mem_range
protected alias ⟨_, iUnion⟩ := absorbs_iUnion | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | _root_.absorbs_iUnion | null |
_root_.Set.Finite.absorbs_biUnion {ι : Type*} {t : ι → Set α} {I : Set ι} (hI : I.Finite) :
Absorbs M s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, Absorbs M s (t i) := by
simp [Absorbs, hI]
protected alias ⟨_, biUnion⟩ := Set.Finite.absorbs_biUnion
@[simp] | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | _root_.Set.Finite.absorbs_biUnion | null |
_root_.absorbs_biUnion_finset {ι : Type*} {t : ι → Set α} {I : Finset ι} :
Absorbs M s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, Absorbs M s (t i) :=
I.finite_toSet.absorbs_biUnion
protected alias ⟨_, biUnion_finset⟩ := absorbs_biUnion_finset | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | _root_.absorbs_biUnion_finset | null |
protected add [AddZeroClass E] [DistribSMul M E]
(h₁ : Absorbs M s₁ t₁) (h₂ : Absorbs M s₂ t₂) : Absorbs M (s₁ + s₂) (t₁ + t₂) :=
h₂.mp <| h₁.eventually.mono fun x hx₁ hx₂ ↦ by rw [smul_add]; exact add_subset_add hx₁ hx₂ | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | add | null |
protected zero [Zero E] [SMulZeroClass M E] {s : Set E} (hs : 0 ∈ s) : Absorbs M s 0 :=
Eventually.of_forall fun _ ↦ zero_subset.2 <| zero_mem_smul_set hs | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | zero | null |
@[simp]
protected Absorbs.univ : Absorbs G₀ univ s :=
(eventually_ne_cobounded 0).mono fun a ha ↦ by rw [smul_set_univ₀ ha]; apply subset_univ | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | Absorbs.univ | null |
absorbs_iff_eventually_cobounded_mapsTo :
Absorbs G₀ s t ↔ ∀ᶠ c in cobounded G₀, MapsTo (c⁻¹ • ·) t s :=
eventually_congr <| (eventually_ne_cobounded 0).mono fun c hc ↦ by
rw [← preimage_smul_inv₀ hc]; rfl
alias ⟨eventually_cobounded_mapsTo, _⟩ := absorbs_iff_eventually_cobounded_mapsTo
@[simp] | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | absorbs_iff_eventually_cobounded_mapsTo | null |
absorbs_inter : Absorbs G₀ (s ∩ t) u ↔ Absorbs G₀ s u ∧ Absorbs G₀ t u := by
simp only [absorbs_iff_eventually_cobounded_mapsTo, mapsTo_inter, eventually_and] | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | absorbs_inter | null |
protected Absorbs.inter (hs : Absorbs G₀ s u) (ht : Absorbs G₀ t u) : Absorbs G₀ (s ∩ t) u :=
absorbs_inter.2 ⟨hs, ht⟩
variable (G₀ u) in | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | Absorbs.inter | null |
Filter.absorbing : Filter α where
sets := {s | Absorbs G₀ s u}
univ_sets := .univ
sets_of_superset h := h.mono_left
inter_sets := .inter
@[simp] | def | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | Filter.absorbing | The filter of sets that absorb `u`. |
Filter.mem_absorbing : s ∈ absorbing G₀ u ↔ Absorbs G₀ s u := .rfl | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | Filter.mem_absorbing | null |
Set.Finite.absorbs_sInter (hS : S.Finite) :
Absorbs G₀ (⋂₀ S) t ↔ ∀ s ∈ S, Absorbs G₀ s t :=
sInter_mem (f := absorbing G₀ t) hS
protected alias ⟨_, Absorbs.sInter⟩ := Set.Finite.absorbs_sInter
@[simp] | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | Set.Finite.absorbs_sInter | null |
absorbs_iInter {ι : Sort*} [Finite ι] {s : ι → Set α} :
Absorbs G₀ (⋂ i, s i) t ↔ ∀ i, Absorbs G₀ (s i) t :=
iInter_mem (f := absorbing G₀ t)
protected alias ⟨_, Absorbs.iInter⟩ := absorbs_iInter | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | absorbs_iInter | null |
Set.Finite.absorbs_biInter {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set α} :
Absorbs G₀ (⋂ i ∈ I, s i) t ↔ ∀ i ∈ I, Absorbs G₀ (s i) t :=
biInter_mem (f := absorbing G₀ t) hI
protected alias ⟨_, Absorbs.biInter⟩ := Set.Finite.absorbs_biInter
@[simp] | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | Set.Finite.absorbs_biInter | null |
absorbs_zero_iff [NeBot (cobounded G₀)]
{E : Type*} [AddMonoid E] [DistribMulAction G₀ E] {s : Set E} :
Absorbs G₀ s 0 ↔ 0 ∈ s := by
simp only [absorbs_iff_eventually_cobounded_mapsTo, ← singleton_zero,
mapsTo_singleton, smul_zero, eventually_const] | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | absorbs_zero_iff | null |
@[simp]
absorbs_neg_neg {s t : Set E} : Absorbs M (-s) (-t) ↔ Absorbs M s t := by simp [Absorbs]
alias ⟨Absorbs.of_neg_neg, Absorbs.neg_neg⟩ := absorbs_neg_neg | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | absorbs_neg_neg | null |
Absorbs.sub {s₁ s₂ t₁ t₂ : Set E} (h₁ : Absorbs M s₁ t₁) (h₂ : Absorbs M s₂ t₂) :
Absorbs M (s₁ - s₂) (t₁ - t₂) := by
simpa only [sub_eq_add_neg] using h₁.add h₂.neg_neg | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | Absorbs.sub | null |
protected mono (ht : Absorbent M s) (hsub : s ⊆ t) : Absorbent M t := fun x ↦
(ht x).mono_left hsub | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | mono | null |
_root_.absorbent_iff_forall_absorbs_singleton : Absorbent M s ↔ ∀ x, Absorbs M s {x} := .rfl | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | _root_.absorbent_iff_forall_absorbs_singleton | null |
protected absorbs (hs : Absorbent M s) {x : α} : Absorbs M s {x} := hs x | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | absorbs | null |
absorbs_finite (hs : Absorbent M s) (ht : t.Finite) : Absorbs M s t := by
rw [← Set.biUnion_of_singleton t]
exact .biUnion ht fun _ _ => hs.absorbs | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | absorbs_finite | null |
vadd_absorbs {M E : Type*} [Bornology M] [AddZeroClass E] [DistribSMul M E]
{s₁ s₂ t : Set E} {x : E} (h₁ : Absorbent M s₁) (h₂ : Absorbs M s₂ t) :
Absorbs M (s₁ + s₂) (x +ᵥ t) := by
rw [← singleton_vadd]; exact (h₁ x).add h₂ | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | vadd_absorbs | null |
absorbent_univ : Absorbent G₀ (univ : Set α) := fun _ ↦ .univ | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | absorbent_univ | null |
absorbent_iff_inv_smul {s : Set α} :
Absorbent G₀ s ↔ ∀ x, ∀ᶠ c in cobounded G₀, c⁻¹ • x ∈ s :=
forall_congr' fun x ↦ by simp only [absorbs_iff_eventually_cobounded_mapsTo, mapsTo_singleton] | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | absorbent_iff_inv_smul | null |
Absorbent.zero_mem [NeBot (cobounded G₀)] [AddMonoid E] [DistribMulAction G₀ E]
{s : Set E} (hs : Absorbent G₀ s) : (0 : E) ∈ s :=
absorbs_zero_iff.1 (hs 0) | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | Absorbent.zero_mem | null |
protected Absorbs.restrict_scalars
{M N α : Type*} [Monoid N] [SMul M N] [SMul M α] [MulAction N α]
[IsScalarTower M N α] [Bornology M] [Bornology N] {s t : Set α} (h : Absorbs N s t)
(hbdd : Tendsto (· • 1 : M → N) (cobounded M) (cobounded N)) :
Absorbs M s t :=
(hbdd.eventually h).mono <| fun x hx ↦ by rwa [smul_one_smul N x s] at hx | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Ring.Action.Pointwise.Set",
"Mathlib.Topology.Bornology.Basic"
] | Mathlib/Topology/Bornology/Absorbs.lean | Absorbs.restrict_scalars | null |
Bornology (α : Type*) where
/-- The filter of cobounded sets in a bornology. -/
cobounded (α) : Filter α
/-- The cobounded filter in a bornology is smaller than the cofinite filter. -/
le_cofinite (α) : cobounded ≤ cofinite
@[deprecated (since := "2025-09-06")] alias Bornology.cobounded' := Bornology.cobounded
@[deprecated (since := "2025-09-06")] alias Bornology.le_cofinite' := Bornology.le_cofinite
@[ext] | class | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | Bornology | A **bornology** on a type `α` is a filter of cobounded sets which contains the cofinite filter.
Such spaces are equivalently specified by their bounded sets, see `Bornology.ofBounded`
and `Bornology.ext_iff_isBounded` |
Bornology.ext (t t' : Bornology α)
(h_cobounded : @Bornology.cobounded α t = @Bornology.cobounded α t') :
t = t' := by
cases t
cases t'
congr | lemma | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | Bornology.ext | null |
@[simps]
Bornology.ofBounded {α : Type*} (B : Set (Set α))
(empty_mem : ∅ ∈ B)
(subset_mem : ∀ s₁ ∈ B, ∀ s₂ ⊆ s₁, s₂ ∈ B)
(union_mem : ∀ s₁ ∈ B, ∀ s₂ ∈ B, s₁ ∪ s₂ ∈ B)
(singleton_mem : ∀ x, {x} ∈ B) : Bornology α where
cobounded := comk (· ∈ B) empty_mem subset_mem union_mem
le_cofinite := by simpa [le_cofinite_iff_compl_singleton_mem] | def | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | Bornology.ofBounded | A constructor for bornologies by specifying the bounded sets,
and showing that they satisfy the appropriate conditions. |
@[simps! cobounded]
Bornology.ofBounded' {α : Type*} (B : Set (Set α))
(empty_mem : ∅ ∈ B)
(subset_mem : ∀ s₁ ∈ B, ∀ s₂ ⊆ s₁, s₂ ∈ B)
(union_mem : ∀ s₁ ∈ B, ∀ s₂ ∈ B, s₁ ∪ s₂ ∈ B)
(sUnion_univ : ⋃₀ B = univ) :
Bornology α :=
Bornology.ofBounded B empty_mem subset_mem union_mem fun x => by
rw [sUnion_eq_univ_iff] at sUnion_univ
rcases sUnion_univ x with ⟨s, hs, hxs⟩
exact subset_mem s hs {x} (singleton_subset_iff.mpr hxs) | def | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | Bornology.ofBounded' | A constructor for bornologies by specifying the bounded sets,
and showing that they satisfy the appropriate conditions. |
IsCobounded [Bornology α] (s : Set α) : Prop :=
s ∈ cobounded α | def | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | IsCobounded | `IsCobounded` is the predicate that `s` is in the filter of cobounded sets in the ambient
bornology on `α` |
IsBounded [Bornology α] (s : Set α) : Prop :=
IsCobounded sᶜ
variable {_ : Bornology α} {s t : Set α} {x : α} | def | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | IsBounded | `IsBounded` is the predicate that `s` is bounded relative to the ambient bornology on `α`. |
isCobounded_def {s : Set α} : IsCobounded s ↔ s ∈ cobounded α :=
Iff.rfl | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isCobounded_def | null |
isBounded_def {s : Set α} : IsBounded s ↔ sᶜ ∈ cobounded α :=
Iff.rfl
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isBounded_def | null |
isBounded_compl_iff : IsBounded sᶜ ↔ IsCobounded s := by
rw [isBounded_def, isCobounded_def, compl_compl]
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isBounded_compl_iff | null |
isCobounded_compl_iff : IsCobounded sᶜ ↔ IsBounded s :=
Iff.rfl
alias ⟨IsBounded.of_compl, IsCobounded.compl⟩ := isBounded_compl_iff
alias ⟨IsCobounded.of_compl, IsBounded.compl⟩ := isCobounded_compl_iff
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isCobounded_compl_iff | null |
isBounded_empty : IsBounded (∅ : Set α) := by
rw [isBounded_def, compl_empty]
exact univ_mem | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isBounded_empty | null |
nonempty_of_not_isBounded (h : ¬IsBounded s) : s.Nonempty := by
rw [nonempty_iff_ne_empty]
rintro rfl
exact h isBounded_empty
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | nonempty_of_not_isBounded | null |
isBounded_singleton : IsBounded ({x} : Set α) := by
rw [isBounded_def]
exact le_cofinite _ (finite_singleton x).compl_mem_cofinite | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isBounded_singleton | null |
isBounded_iff_forall_mem : IsBounded s ↔ ∀ x ∈ s, IsBounded s :=
⟨fun h _ _ ↦ h, fun h ↦ by
rcases s.eq_empty_or_nonempty with rfl | ⟨x, hx⟩
exacts [isBounded_empty, h x hx]⟩
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isBounded_iff_forall_mem | null |
isCobounded_univ : IsCobounded (univ : Set α) :=
univ_mem
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isCobounded_univ | null |
isCobounded_inter : IsCobounded (s ∩ t) ↔ IsCobounded s ∧ IsCobounded t :=
inter_mem_iff | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isCobounded_inter | null |
IsCobounded.inter (hs : IsCobounded s) (ht : IsCobounded t) : IsCobounded (s ∩ t) :=
isCobounded_inter.2 ⟨hs, ht⟩
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | IsCobounded.inter | null |
isBounded_union : IsBounded (s ∪ t) ↔ IsBounded s ∧ IsBounded t := by
simp only [← isCobounded_compl_iff, compl_union, isCobounded_inter] | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isBounded_union | null |
IsBounded.union (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s ∪ t) :=
isBounded_union.2 ⟨hs, ht⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | IsBounded.union | null |
IsCobounded.superset (hs : IsCobounded s) (ht : s ⊆ t) : IsCobounded t :=
mem_of_superset hs ht | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | IsCobounded.superset | null |
IsBounded.subset (ht : IsBounded t) (hs : s ⊆ t) : IsBounded s :=
ht.superset (compl_subset_compl.mpr hs)
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | IsBounded.subset | null |
sUnion_bounded_univ : ⋃₀ { s : Set α | IsBounded s } = univ :=
sUnion_eq_univ_iff.2 fun a => ⟨{a}, isBounded_singleton, mem_singleton a⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | sUnion_bounded_univ | null |
IsBounded.insert (h : IsBounded s) (x : α) : IsBounded (insert x s) :=
isBounded_singleton.union h
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | IsBounded.insert | null |
isBounded_insert : IsBounded (insert x s) ↔ IsBounded s :=
⟨fun h ↦ h.subset (subset_insert _ _), (.insert · x)⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isBounded_insert | null |
comap_cobounded_le_iff [Bornology β] {f : α → β} :
(cobounded β).comap f ≤ cobounded α ↔ ∀ ⦃s⦄, IsBounded s → IsBounded (f '' s) := by
refine
⟨fun h s hs => ?_, fun h t ht =>
⟨(f '' tᶜ)ᶜ, h <| IsCobounded.compl ht, compl_subset_comm.1 <| subset_preimage_image _ _⟩⟩
obtain ⟨t, ht, hts⟩ := h hs.compl
rw [subset_compl_comm, ← preimage_compl] at hts
exact (IsCobounded.compl ht).subset ((image_mono hts).trans <| image_preimage_subset _ _) | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | comap_cobounded_le_iff | null |
ext_iff' {t t' : Bornology α} :
t = t' ↔ ∀ s, s ∈ @cobounded α t ↔ s ∈ @cobounded α t' :=
Bornology.ext_iff.trans Filter.ext_iff | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | ext_iff' | null |
ext_iff_isBounded {t t' : Bornology α} :
t = t' ↔ ∀ s, @IsBounded α t s ↔ @IsBounded α t' s :=
ext_iff'.trans compl_surjective.forall
variable {s : Set α} | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | ext_iff_isBounded | null |
isCobounded_ofBounded_iff (B : Set (Set α)) {empty_mem subset_mem union_mem sUnion_univ} :
@IsCobounded _ (ofBounded B empty_mem subset_mem union_mem sUnion_univ) s ↔ sᶜ ∈ B :=
Iff.rfl | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isCobounded_ofBounded_iff | null |
isBounded_ofBounded_iff (B : Set (Set α)) {empty_mem subset_mem union_mem sUnion_univ} :
@IsBounded _ (ofBounded B empty_mem subset_mem union_mem sUnion_univ) s ↔ s ∈ B := by
rw [isBounded_def, ofBounded_cobounded, compl_mem_comk]
variable [Bornology α] | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isBounded_ofBounded_iff | null |
isCobounded_biInter {s : Set ι} {f : ι → Set α} (hs : s.Finite) :
IsCobounded (⋂ i ∈ s, f i) ↔ ∀ i ∈ s, IsCobounded (f i) :=
biInter_mem hs
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isCobounded_biInter | null |
isCobounded_biInter_finset (s : Finset ι) {f : ι → Set α} :
IsCobounded (⋂ i ∈ s, f i) ↔ ∀ i ∈ s, IsCobounded (f i) :=
biInter_finset_mem s
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isCobounded_biInter_finset | null |
isCobounded_iInter [Finite ι] {f : ι → Set α} :
IsCobounded (⋂ i, f i) ↔ ∀ i, IsCobounded (f i) :=
iInter_mem | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isCobounded_iInter | null |
isCobounded_sInter {S : Set (Set α)} (hs : S.Finite) :
IsCobounded (⋂₀ S) ↔ ∀ s ∈ S, IsCobounded s :=
sInter_mem hs | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isCobounded_sInter | null |
isBounded_biUnion {s : Set ι} {f : ι → Set α} (hs : s.Finite) :
IsBounded (⋃ i ∈ s, f i) ↔ ∀ i ∈ s, IsBounded (f i) := by
simp only [← isCobounded_compl_iff, compl_iUnion, isCobounded_biInter hs] | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isBounded_biUnion | null |
isBounded_biUnion_finset (s : Finset ι) {f : ι → Set α} :
IsBounded (⋃ i ∈ s, f i) ↔ ∀ i ∈ s, IsBounded (f i) :=
isBounded_biUnion s.finite_toSet | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isBounded_biUnion_finset | null |
isBounded_sUnion {S : Set (Set α)} (hs : S.Finite) :
IsBounded (⋃₀ S) ↔ ∀ s ∈ S, IsBounded s := by rw [sUnion_eq_biUnion, isBounded_biUnion hs]
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isBounded_sUnion | null |
isBounded_iUnion [Finite ι] {s : ι → Set α} :
IsBounded (⋃ i, s i) ↔ ∀ i, IsBounded (s i) := by
rw [← sUnion_range, isBounded_sUnion (finite_range s), forall_mem_range] | theorem | Topology | [
"Mathlib.Order.Filter.Cofinite"
] | Mathlib/Topology/Bornology/Basic.lean | isBounded_iUnion | null |
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