# Documentation

Mathlib.Topology.Bornology.Constructions

# Bornology structure on products and subtypes #

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

instance instBornologyProd {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] :
Bornology (α × β)
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• One or more equations did not get rendered due to their size.
instance instBornologyForAll {ι : Type u_1} {π : ιType u_2} [inst : ] [inst : (i : ι) → Bornology (π i)] :
Bornology ((i : ι) → π i)
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def Bornology.induced {α : Type u_1} {β : Type u_2} [inst : ] (f : αβ) :

Inverse image of a bornology.

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• = { cobounded' := , le_cofinite' := (_ : Filter.cofinite) }
instance instBornologySubtype {α : Type u_1} [inst : ] {p : αProp} :
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### Bounded sets in α × β× β#

theorem Bornology.cobounded_prod {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] :
theorem Bornology.isBounded_image_fst_and_snd {α : Type u_2} {β : Type u_1} [inst : ] [inst : ] {s : Set (α × β)} :
Bornology.IsBounded (Prod.fst '' s) Bornology.IsBounded (Prod.snd '' s)
theorem Bornology.IsBounded.fst_of_prod {α : Type u_2} {β : Type u_1} [inst : ] [inst : ] {s : Set α} {t : Set β} (h : Bornology.IsBounded (s ×ˢ t)) (ht : ) :
theorem Bornology.IsBounded.snd_of_prod {α : Type u_2} {β : Type u_1} [inst : ] [inst : ] {s : Set α} {t : Set β} (h : Bornology.IsBounded (s ×ˢ t)) (hs : ) :
theorem Bornology.IsBounded.prod {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] {s : Set α} {t : Set β} (hs : ) (ht : ) :
theorem Bornology.isBounded_prod_of_nonempty {α : Type u_2} {β : Type u_1} [inst : ] [inst : ] {s : Set α} {t : Set β} (hne : Set.Nonempty (s ×ˢ t)) :
theorem Bornology.isBounded_prod {α : Type u_2} {β : Type u_1} [inst : ] [inst : ] {s : Set α} {t : Set β} :
theorem Bornology.isBounded_prod_self {α : Type u_1} [inst : ] {s : Set α} :

### Bounded sets in Π i, π i#

theorem Bornology.cobounded_pi {ι : Type u_1} {π : ιType u_2} [inst : ] [inst : (i : ι) → Bornology (π i)] :
Bornology.cobounded ((i : ι) → π i) = Filter.coprodᵢ fun i => Bornology.cobounded (π i)
theorem Bornology.forall_isBounded_image_eval_iff {ι : Type u_1} {π : ιType u_2} [inst : ] [inst : (i : ι) → Bornology (π i)] {s : Set ((i : ι) → π i)} :
(∀ (i : ι), )
theorem Bornology.IsBounded.pi {ι : Type u_2} {π : ιType u_1} [inst : ] [inst : (i : ι) → Bornology (π i)] {S : (i : ι) → Set (π i)} (h : ∀ (i : ι), Bornology.IsBounded (S i)) :
theorem Bornology.isBounded_pi_of_nonempty {ι : Type u_1} {π : ιType u_2} [inst : ] [inst : (i : ι) → Bornology (π i)] {S : (i : ι) → Set (π i)} (hne : Set.Nonempty (Set.pi Set.univ S)) :
Bornology.IsBounded (Set.pi Set.univ S) ∀ (i : ι), Bornology.IsBounded (S i)
theorem Bornology.isBounded_pi {ι : Type u_2} {π : ιType u_1} [inst : ] [inst : (i : ι) → Bornology (π i)] {S : (i : ι) → Set (π i)} :
Bornology.IsBounded (Set.pi Set.univ S) (i, S i = ) ∀ (i : ι), Bornology.IsBounded (S i)

### Bounded sets in {x // p x}#

theorem Bornology.isBounded_induced {α : Type u_1} {β : Type u_2} [inst : ] {f : αβ} {s : Set α} :
theorem Bornology.isBounded_image_subtype_val {α : Type u_1} [inst : ] {p : αProp} {s : Set { x // p x }} :
Bornology.IsBounded (Subtype.val '' s)

### Bounded spaces #

instance instBoundedSpaceProdInstBornologyProd {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] [inst : ] [inst : ] :
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instance instBoundedSpaceForAllInstBornologyForAll {ι : Type u_1} {π : ιType u_2} [inst : ] [inst : (i : ι) → Bornology (π i)] [inst : ∀ (i : ι), BoundedSpace (π i)] :
BoundedSpace ((i : ι) → π i)
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theorem boundedSpace_induced_iff {α : Type u_1} {β : Type u_2} [inst : ] {f : αβ} :
theorem boundedSpace_subtype_iff {α : Type u_1} [inst : ] {p : αProp} :
theorem boundedSpace_val_set_iff {α : Type u_1} [inst : ] {s : Set α} :
theorem Bornology.IsBounded.boundedSpace_subtype {α : Type u_1} [inst : ] {p : αProp} :
Bornology.IsBounded { x | p x }

Alias of the reverse direction of boundedSpace_subtype_iff.

theorem Bornology.IsBounded.boundedSpace_val {α : Type u_1} [inst : ] {s : Set α} :

Alias of the reverse direction of boundedSpace_val_set_iff.

instance instBoundedSpaceSubtypeInstBornologySubtype {α : Type u_1} [inst : ] [inst : ] {p : αProp} :
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### Additive, Multiplicative#

The bornology on those type synonyms is inherited without change.

instance instBornologyAdditive {α : Type u_1} [inst : ] :
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instance instBornologyMultiplicative {α : Type u_1} [inst : ] :
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• instBornologyMultiplicative = inst
instance instBoundedSpaceAdditiveInstBornologyAdditive {α : Type u_1} [inst : ] [inst : ] :
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instance instBoundedSpaceMultiplicativeInstBornologyMultiplicative {α : Type u_1} [inst : ] [inst : ] :
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### Order dual #

The bornology on this type synonym is inherited without change.

instance instBornologyOrderDual {α : Type u_1} [inst : ] :
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• instBornologyOrderDual = inst
instance instBoundedSpaceOrderDualInstBornologyOrderDual {α : Type u_1} [inst : ] [inst : ] :
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