Basic theory of bornology

We develop the basic theory of bornologies. Instead of axiomatizing bounded sets and defining bornologies in terms of those, we recognize that the cobounded sets form a filter and define a bornology as a filter of cobounded sets which contains the cofinite filter. This allows us to make use of the extensive library for filters, but we also provide the relevant connecting results for bounded sets.

The specification of a bornology in terms of the cobounded filter is equivalent to the standard one (e.g., see [Bourbaki, Topological Vector Spaces][bourbaki1987], covering bornology, now often called simply bornology) in terms of bounded sets (see Bornology.ofBounded, IsBounded.union, IsBounded.subset), except that we do not allow the empty bornology (that is, we require that some set must be bounded; equivalently, ∅ is bounded). In the literature the cobounded filter is generally referred to as the filter at infinity.

Main definitions

• Bornology α: a class consisting of cobounded : Filter α and a proof that this filter contains the cofinite filter.
• Bornology.IsCobounded: the predicate that a set is a member of the cobounded α filter. For s : Set α, one should prefer Bornology.IsCobounded s over s ∈ cobounded α.
• bornology.IsBounded: the predicate that states a set is bounded (i.e., the complement of a cobounded set). One should prefer Bornology.IsBounded s over sᶜ ∈ cobounded α.
• BoundedSpace α: a class extending Bornology α with the condition Bornology.IsBounded (Set.univ : Set α)

Although use of cobounded α is discouraged for indicating the (co)boundedness of individual sets, it is intended for regular use as a filter on α.

class Bornology (α : Type u_4) : Type u_4

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

• cobounded' : Filter α

  The filter of cobounded sets in a bornology. This is a field of the structure, but one should always prefer Bornology.cobounded because it makes the α argument explicit.

• le_cofinite' : Bornology.cobounded' ≤ Filter.cofinite

  The cobounded filter in a bornology is smaller than the cofinite filter. This is a field of the structure, but one should always prefer Bornology.le_cofinite because it makes the α argument explicit.

theorem Bornology.le_cofinite' {α : Type u_4} [self : Bornology α] : Bornology.cobounded' ≤ Filter.cofinite

The cobounded filter in a bornology is smaller than the cofinite filter. This is a field of the structure, but one should always prefer Bornology.le_cofinite because it makes the α argument explicit.

def Bornology.cobounded (α : Type u_4) [Bornology α] : Filter α

The filter of cobounded sets in a bornology.

Equations

• Bornology.cobounded α = Bornology.cobounded'

def Bornology.Simps.cobounded (α : Type u_4) [Bornology α] : Filter α

Alias of Bornology.cobounded.

---

The filter of cobounded sets in a bornology.

Equations

• Bornology.Simps.cobounded = Bornology.cobounded

theorem Bornology.le_cofinite (α : Type u_4) [Bornology α] : Bornology.cobounded α ≤ Filter.cofinite

theorem Bornology.ext_iff {α : Type u_2} {t : Bornology α} {t' : Bornology α} : t = t' ↔ Bornology.cobounded α = Bornology.cobounded α

theorem Bornology.ext {α : Type u_2} (t : Bornology α) (t' : Bornology α) (h_cobounded : Bornology.cobounded α = Bornology.cobounded α) : t = t'

@[simp]

theorem Bornology.ofBounded_cobounded {α : Type u_4} (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 : α), {x} ∈ B) : Bornology.cobounded α = Filter.comk (fun (x : Set α) => x ∈ B) empty_mem subset_mem union_mem

def Bornology.ofBounded {α : Type u_4} (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 : α), {x} ∈ B) : Bornology α

A constructor for bornologies by specifying the bounded sets, and showing that they satisfy the appropriate conditions.

Equations

• Bornology.ofBounded B empty_mem subset_mem union_mem singleton_mem = { cobounded' := Filter.comk (fun (x : Set α) => x ∈ B) empty_mem subset_mem union_mem, le_cofinite' := ⋯ }

@[simp]

theorem Bornology.ofBounded'_cobounded {α : Type u_4} (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 = Set.univ) : Bornology.cobounded α = Filter.comk (fun (x : Set α) => x ∈ B) empty_mem subset_mem union_mem

def Bornology.ofBounded' {α : Type u_4} (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 = Set.univ) : Bornology α

A constructor for bornologies by specifying the bounded sets, and showing that they satisfy the appropriate conditions.

Equations

• Bornology.ofBounded' B empty_mem subset_mem union_mem sUnion_univ = Bornology.ofBounded B empty_mem subset_mem union_mem ⋯

def Bornology.IsCobounded {α : Type u_2} [Bornology α] (s : Set α) : Prop

IsCobounded is the predicate that s is in the filter of cobounded sets in the ambient bornology on α

Equations

• Bornology.IsCobounded s = (s ∈ Bornology.cobounded α)

def Bornology.IsBounded {α : Type u_2} [Bornology α] (s : Set α) : Prop

IsBounded is the predicate that s is bounded relative to the ambient bornology on α.

Equations

• Bornology.IsBounded s = Bornology.IsCobounded sᶜ

theorem Bornology.isCobounded_def {α : Type u_2} : ∀ {x : Bornology α} {s : Set α}, Bornology.IsCobounded s ↔ s ∈ Bornology.cobounded α

theorem Bornology.isBounded_def {α : Type u_2} : ∀ {x : Bornology α} {s : Set α}, Bornology.IsBounded s ↔ sᶜ ∈ Bornology.cobounded α

@[simp]

theorem Bornology.isBounded_compl_iff {α : Type u_2} : ∀ {x : Bornology α} {s : Set α}, Bornology.IsBounded sᶜ ↔ Bornology.IsCobounded s

@[simp]

theorem Bornology.isCobounded_compl_iff {α : Type u_2} : ∀ {x : Bornology α} {s : Set α}, Bornology.IsCobounded sᶜ ↔ Bornology.IsBounded s

theorem Bornology.IsCobounded.compl {α : Type u_2} : ∀ {x : Bornology α} {s : Set α}, Bornology.IsCobounded s → Bornology.IsBounded sᶜ

Alias of the reverse direction of Bornology.isBounded_compl_iff.

theorem Bornology.IsBounded.of_compl {α : Type u_2} : ∀ {x : Bornology α} {s : Set α}, Bornology.IsBounded sᶜ → Bornology.IsCobounded s

Alias of the forward direction of Bornology.isBounded_compl_iff.

theorem Bornology.IsCobounded.of_compl {α : Type u_2} : ∀ {x : Bornology α} {s : Set α}, Bornology.IsCobounded sᶜ → Bornology.IsBounded s

Alias of the forward direction of Bornology.isCobounded_compl_iff.

theorem Bornology.IsBounded.compl {α : Type u_2} : ∀ {x : Bornology α} {s : Set α}, Bornology.IsBounded s → Bornology.IsCobounded sᶜ

Alias of the reverse direction of Bornology.isCobounded_compl_iff.

@[simp]

theorem Bornology.isBounded_empty {α : Type u_2} : ∀ {x : Bornology α}, Bornology.IsBounded ∅

theorem Bornology.nonempty_of_not_isBounded {α : Type u_2} : ∀ {x : Bornology α} {s : Set α}, ¬Bornology.IsBounded s → s.Nonempty

@[simp]

theorem Bornology.isBounded_singleton {α : Type u_2} : ∀ {x : Bornology α} {x_1 : α}, Bornology.IsBounded {x_1}

theorem Bornology.isBounded_iff_forall_mem {α : Type u_2} : ∀ {x : Bornology α} {s : Set α}, Bornology.IsBounded s ↔ ∀ x_1 ∈ s, Bornology.IsBounded s

@[simp]

theorem Bornology.isCobounded_univ {α : Type u_2} : ∀ {x : Bornology α}, Bornology.IsCobounded Set.univ

@[simp]

theorem Bornology.isCobounded_inter {α : Type u_2} : ∀ {x : Bornology α} {s t : Set α}, Bornology.IsCobounded (s ∩ t) ↔ Bornology.IsCobounded s ∧ Bornology.IsCobounded t

theorem Bornology.IsCobounded.inter {α : Type u_2} : ∀ {x : Bornology α} {s t : Set α}, Bornology.IsCobounded s → Bornology.IsCobounded t → Bornology.IsCobounded (s ∩ t)

@[simp]

theorem Bornology.isBounded_union {α : Type u_2} : ∀ {x : Bornology α} {s t : Set α}, Bornology.IsBounded (s ∪ t) ↔ Bornology.IsBounded s ∧ Bornology.IsBounded t

theorem Bornology.IsBounded.union {α : Type u_2} : ∀ {x : Bornology α} {s t : Set α}, Bornology.IsBounded s → Bornology.IsBounded t → Bornology.IsBounded (s ∪ t)

theorem Bornology.IsCobounded.superset {α : Type u_2} : ∀ {x : Bornology α} {s t : Set α}, Bornology.IsCobounded s → s ⊆ t → Bornology.IsCobounded t

theorem Bornology.IsBounded.subset {α : Type u_2} : ∀ {x : Bornology α} {s t : Set α}, Bornology.IsBounded t → s ⊆ t → Bornology.IsBounded s

@[simp]

theorem Bornology.sUnion_bounded_univ {α : Type u_2} : ∀ {x : Bornology α}, ⋃₀ {s : Set α | Bornology.IsBounded s} = Set.univ

theorem Bornology.IsBounded.insert {α : Type u_2} : ∀ {x : Bornology α} {s : Set α}, Bornology.IsBounded s → ∀ (x_1 : α), Bornology.IsBounded (insert x_1 s)

@[simp]

theorem Bornology.isBounded_insert {α : Type u_2} : ∀ {x : Bornology α} {s : Set α} {x_1 : α}, Bornology.IsBounded (insert x_1 s) ↔ Bornology.IsBounded s

theorem Bornology.comap_cobounded_le_iff {α : Type u_2} {β : Type u_3} : ∀ {x : Bornology α} [inst : Bornology β] {f : α → β}, Filter.comap f (Bornology.cobounded β) ≤ Bornology.cobounded α ↔ ∀ ⦃s : Set α⦄, Bornology.IsBounded s → Bornology.IsBounded (f '' s)

theorem Bornology.ext_iff' {α : Type u_2} {t : Bornology α} {t' : Bornology α} : t = t' ↔ ∀ (s : Set α), s ∈ Bornology.cobounded α ↔ s ∈ Bornology.cobounded α

theorem Bornology.ext_iff_isBounded {α : Type u_2} {t : Bornology α} {t' : Bornology α} : t = t' ↔ ∀ (s : Set α), Bornology.IsBounded s ↔ Bornology.IsBounded s

theorem Bornology.isCobounded_ofBounded_iff {α : Type u_2} {s : Set α} (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 : ∀ (x : α), {x} ∈ B} : Bornology.IsCobounded s ↔ sᶜ ∈ B

theorem Bornology.isBounded_ofBounded_iff {α : Type u_2} {s : Set α} (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 : ∀ (x : α), {x} ∈ B} : Bornology.IsBounded s ↔ s ∈ B

theorem Bornology.isCobounded_biInter {ι : Type u_1} {α : Type u_2} [Bornology α] {s : Set ι} {f : ι → Set α} (hs : s.Finite) : Bornology.IsCobounded (⋂ i ∈ s, f i) ↔ ∀ i ∈ s, Bornology.IsCobounded (f i)

@[simp]

theorem Bornology.isCobounded_biInter_finset {ι : Type u_1} {α : Type u_2} [Bornology α] (s : Finset ι) {f : ι → Set α} : Bornology.IsCobounded (⋂ i ∈ s, f i) ↔ ∀ i ∈ s, Bornology.IsCobounded (f i)

@[simp]

theorem Bornology.isCobounded_iInter {ι : Type u_1} {α : Type u_2} [Bornology α] [Finite ι] {f : ι → Set α} : Bornology.IsCobounded (⋂ (i : ι), f i) ↔ ∀ (i : ι), Bornology.IsCobounded (f i)

theorem Bornology.isCobounded_sInter {α : Type u_2} [Bornology α] {S : Set (Set α)} (hs : S.Finite) : Bornology.IsCobounded (⋂₀ S) ↔ ∀ s ∈ S, Bornology.IsCobounded s

theorem Bornology.isBounded_biUnion {ι : Type u_1} {α : Type u_2} [Bornology α] {s : Set ι} {f : ι → Set α} (hs : s.Finite) : Bornology.IsBounded (⋃ i ∈ s, f i) ↔ ∀ i ∈ s, Bornology.IsBounded (f i)

theorem Bornology.isBounded_biUnion_finset {ι : Type u_1} {α : Type u_2} [Bornology α] (s : Finset ι) {f : ι → Set α} : Bornology.IsBounded (⋃ i ∈ s, f i) ↔ ∀ i ∈ s, Bornology.IsBounded (f i)

theorem Bornology.isBounded_sUnion {α : Type u_2} [Bornology α] {S : Set (Set α)} (hs : S.Finite) : Bornology.IsBounded (⋃₀ S) ↔ ∀ s ∈ S, Bornology.IsBounded s

@[simp]

theorem Bornology.isBounded_iUnion {ι : Type u_1} {α : Type u_2} [Bornology α] [Finite ι] {s : ι → Set α} : Bornology.IsBounded (⋃ (i : ι), s i) ↔ ∀ (i : ι), Bornology.IsBounded (s i)

theorem Bornology.eventually_ne_cobounded {α : Type u_2} [Bornology α] (a : α) : ∀ᶠ (x : α) in Bornology.cobounded α, x ≠ a

theorem Filter.HasBasis.disjoint_cobounded_iff {α : Type u_2} [Bornology α] {ι : Sort u_4} {p : ι → Prop} {s : ι → Set α} {l : Filter α} (h : l.HasBasis p s) : Disjoint l (Bornology.cobounded α) ↔ ∃ (i : ι), p i ∧ Bornology.IsBounded (s i)

theorem Set.Finite.isBounded {α : Type u_2} [Bornology α] {s : Set α} (hs : s.Finite) : Bornology.IsBounded s

theorem Filter.Tendsto.eventually_ne_cobounded {α : Type u_2} {β : Type u_3} [Bornology α] {f : β → α} {l : Filter β} (h : Filter.Tendsto f l (Bornology.cobounded α)) (a : α) : ∀ᶠ (x : β) in l, f x ≠ a

instance instBornologyPUnit : Bornology PUnit.{u_4 + 1}

Equations

• instBornologyPUnit = { cobounded' := ⊥, le_cofinite' := instBornologyPUnit.proof_1 }

@[reducible, inline]

abbrev Bornology.cofinite {α : Type u_2} : Bornology α

The cofinite filter as a bornology

Equations

• Bornology.cofinite = { cobounded' := Filter.cofinite, le_cofinite' := ⋯ }

class BoundedSpace (α : Type u_4) [Bornology α] : Prop

A space with a Bornology is a bounded space if Set.univ : Set α is bounded.

• bounded_univ : Bornology.IsBounded Set.univ

  The Set.univ is bounded.

theorem BoundedSpace.bounded_univ {α : Type u_4} : ∀ {inst : Bornology α} [self : BoundedSpace α], Bornology.IsBounded Set.univ

The Set.univ is bounded.

@[instance 100]

instance BoundedSpace.of_finite {α : Type u_4} [Bornology α] [Finite α] : BoundedSpace α

A finite space is bounded.

Equations

• ⋯ = ⋯

theorem Bornology.isBounded_univ {α : Type u_2} [Bornology α] : Bornology.IsBounded Set.univ ↔ BoundedSpace α

theorem Bornology.cobounded_eq_bot_iff {α : Type u_2} [Bornology α] : Bornology.cobounded α = ⊥ ↔ BoundedSpace α

theorem Bornology.IsBounded.all {α : Type u_2} [Bornology α] [BoundedSpace α] (s : Set α) : Bornology.IsBounded s

theorem Bornology.IsCobounded.all {α : Type u_2} [Bornology α] [BoundedSpace α] (s : Set α) : Bornology.IsCobounded s

@[simp]

theorem Bornology.cobounded_eq_bot (α : Type u_2) [Bornology α] [BoundedSpace α] : Bornology.cobounded α = ⊥

instance OrderDual.instBornology {α : Type u_2} [Bornology α] : Bornology αᵒᵈ

Equations

• OrderDual.instBornology = inst

@[simp]

theorem OrderDual.isCobounded_preimage_ofDual {α : Type u_2} [Bornology α] {s : Set α} : Bornology.IsCobounded (⇑OrderDual.ofDual ⁻¹' s) ↔ Bornology.IsCobounded s

@[simp]

theorem OrderDual.isCobounded_preimage_toDual {α : Type u_2} [Bornology α] {s : Set αᵒᵈ} : Bornology.IsCobounded (⇑OrderDual.toDual ⁻¹' s) ↔ Bornology.IsCobounded s

@[simp]

theorem OrderDual.isBounded_preimage_ofDual {α : Type u_2} [Bornology α] {s : Set α} : Bornology.IsBounded (⇑OrderDual.ofDual ⁻¹' s) ↔ Bornology.IsBounded s

@[simp]

theorem OrderDual.isBounded_preimage_toDual {α : Type u_2} [Bornology α] {s : Set αᵒᵈ} : Bornology.IsBounded (⇑OrderDual.toDual ⁻¹' s) ↔ Bornology.IsBounded s