Bornology structure on products and subtypes

In this file we define Bornology and BoundedSpace instances on α × β, Π i, X i, and {x // p x}. We also prove basic lemmas about Bornology.cobounded and Bornology.IsBounded on these types.

@[instance_reducible]

instance Prod.instBornology {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] : Bornology (α × β)

Equations

• Prod.instBornology = { cobounded := (Bornology.cobounded α).coprod (Bornology.cobounded β), le_cofinite := ⋯ }

@[instance_reducible]

instance Pi.instBornology {ι : Type u_3} {X : ι → Type u_4} [(i : ι) → Bornology (X i)] : Bornology ((i : ι) → X i)

Equations

• Pi.instBornology = { cobounded := Filter.coprodᵢ fun (i : ι) => Bornology.cobounded (X i), le_cofinite := ⋯ }

@[reducible, inline]

abbrev Bornology.induced {α : Type u_5} {β : Type u_6} [Bornology β] (f : α → β) : Bornology α

Inverse image of a bornology.

Equations

• Bornology.induced f = { cobounded := Filter.comap f (Bornology.cobounded β), le_cofinite := ⋯ }

@[instance_reducible]

instance instBornologySubtype {α : Type u_1} [Bornology α] {p : α → Prop} : Bornology (Subtype p)

Equations

• instBornologySubtype = Bornology.induced Subtype.val

Bounded sets in α × β

theorem Bornology.cobounded_prod {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] : cobounded (α × β) = (cobounded α).coprod (cobounded β)

theorem Bornology.isBounded_image_fst_and_snd {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set (α × β)} : IsBounded (Prod.fst '' s) ∧ IsBounded (Prod.snd '' s) ↔ IsBounded s

theorem Bornology.IsBounded.image_fst {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set (α × β)} (hs : IsBounded s) : IsBounded (Prod.fst '' s)

theorem Bornology.IsBounded.image_snd {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set (α × β)} (hs : IsBounded s) : IsBounded (Prod.snd '' s)

theorem Bornology.IsBounded.fst_of_prod {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set α} {t : Set β} (h : IsBounded (s ×ˢ t)) (ht : t.Nonempty) : IsBounded s

theorem Bornology.IsBounded.snd_of_prod {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set α} {t : Set β} (h : IsBounded (s ×ˢ t)) (hs : s.Nonempty) : IsBounded t

theorem Bornology.IsBounded.prod {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set α} {t : Set β} (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s ×ˢ t)

theorem Bornology.isBounded_prod_of_nonempty {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set α} {t : Set β} (hne : (s ×ˢ t).Nonempty) : IsBounded (s ×ˢ t) ↔ IsBounded s ∧ IsBounded t

theorem Bornology.isBounded_prod {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set α} {t : Set β} : IsBounded (s ×ˢ t) ↔ s = ∅ ∨ t = ∅ ∨ IsBounded s ∧ IsBounded t

theorem Bornology.isBounded_prod_self {α : Type u_1} [Bornology α] {s : Set α} : IsBounded (s ×ˢ s) ↔ IsBounded s

Bounded sets in Π i, X i

theorem Bornology.cobounded_pi {ι : Type u_3} {X : ι → Type u_4} [(i : ι) → Bornology (X i)] : cobounded ((i : ι) → X i) = Filter.coprodᵢ fun (i : ι) => cobounded (X i)

theorem Bornology.forall_isBounded_image_eval_iff {ι : Type u_3} {X : ι → Type u_4} [(i : ι) → Bornology (X i)] {s : Set ((i : ι) → X i)} : (∀ (i : ι), IsBounded (Function.eval i '' s)) ↔ IsBounded s

theorem Bornology.IsBounded.image_eval {ι : Type u_3} {X : ι → Type u_4} [(i : ι) → Bornology (X i)] {s : Set ((i : ι) → X i)} (hs : IsBounded s) (i : ι) : IsBounded (Function.eval i '' s)

theorem Bornology.IsBounded.pi {ι : Type u_3} {X : ι → Type u_4} [(i : ι) → Bornology (X i)] {S : (i : ι) → Set (X i)} (h : ∀ (i : ι), IsBounded (S i)) : IsBounded (Set.univ.pi S)

theorem Bornology.isBounded_pi_of_nonempty {ι : Type u_3} {X : ι → Type u_4} [(i : ι) → Bornology (X i)] {S : (i : ι) → Set (X i)} (hne : (Set.univ.pi S).Nonempty) : IsBounded (Set.univ.pi S) ↔ ∀ (i : ι), IsBounded (S i)

theorem Bornology.isBounded_pi {ι : Type u_3} {X : ι → Type u_4} [(i : ι) → Bornology (X i)] {S : (i : ι) → Set (X i)} : IsBounded (Set.univ.pi S) ↔ (∃ (i : ι), S i = ∅) ∨ ∀ (i : ι), IsBounded (S i)

Bounded sets in {x // p x}

theorem Bornology.isBounded_induced {α : Type u_5} {β : Type u_6} [Bornology β] {f : α → β} {s : Set α} : IsBounded s ↔ IsBounded (f '' s)

theorem Bornology.isBounded_image_subtype_val {α : Type u_1} [Bornology α] {p : α → Prop} {s : Set { x : α // p x }} : IsBounded (Subtype.val '' s) ↔ IsBounded s

Bounded spaces

instance instBoundedSpaceProd {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] [BoundedSpace α] [BoundedSpace β] : BoundedSpace (α × β)

instance instBoundedSpaceForall {ι : Type u_3} {X : ι → Type u_4} [(i : ι) → Bornology (X i)] [∀ (i : ι), BoundedSpace (X i)] : BoundedSpace ((i : ι) → X i)

theorem boundedSpace_induced_iff {α : Type u_5} {β : Type u_6} [Bornology β] {f : α → β} : BoundedSpace α ↔ Bornology.IsBounded (Set.range f)

theorem boundedSpace_subtype_iff {α : Type u_1} [Bornology α] {p : α → Prop} : BoundedSpace (Subtype p) ↔ Bornology.IsBounded {x : α | p x}

theorem boundedSpace_val_set_iff {α : Type u_1} [Bornology α] {s : Set α} : BoundedSpace ↑s ↔ Bornology.IsBounded s

theorem Bornology.IsBounded.boundedSpace_subtype {α : Type u_1} [Bornology α] {p : α → Prop} : IsBounded {x : α | p x} → BoundedSpace (Subtype p)

Alias of the reverse direction of boundedSpace_subtype_iff.

theorem Bornology.IsBounded.boundedSpace_val {α : Type u_1} [Bornology α] {s : Set α} : IsBounded s → BoundedSpace ↑s

Alias of the reverse direction of boundedSpace_val_set_iff.

instance instBoundedSpaceSubtype {α : Type u_1} [Bornology α] [BoundedSpace α] {p : α → Prop} : BoundedSpace (Subtype p)

Additive, Multiplicative

The bornology on those type synonyms is inherited without change.

@[instance_reducible]

instance instBornologyAdditive {α : Type u_1} [Bornology α] : Bornology (Additive α)

Equations

• instBornologyAdditive = inst✝

@[instance_reducible]

instance instBornologyMultiplicative {α : Type u_1} [Bornology α] : Bornology (Multiplicative α)

Equations

• instBornologyMultiplicative = inst✝

instance instBoundedSpaceAdditive {α : Type u_1} [Bornology α] [BoundedSpace α] : BoundedSpace (Additive α)

instance instBoundedSpaceMultiplicative {α : Type u_1} [Bornology α] [BoundedSpace α] : BoundedSpace (Multiplicative α)

Order dual

The bornology on this type synonym is inherited without change.

@[instance_reducible]

instance instBornologyOrderDual {α : Type u_1} [Bornology α] : Bornology αᵒᵈ

Equations

• instBornologyOrderDual = inst✝

instance instBoundedSpaceOrderDual {α : Type u_1} [Bornology α] [BoundedSpace α] : BoundedSpace αᵒᵈ