Documentation

Mathlib.Topology.Bornology.Constructions

Bornology structure on products and subtypes #

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

instance Prod.instBornology {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] :
Bornology (α × β)
Equations
instance Pi.instBornology {ι : Type u_3} {π : ιType u_4} [(i : ι) → Bornology (π i)] :
Bornology ((i : ι) → π i)
Equations
@[reducible]
def Bornology.induced {α : Type u_5} {β : Type u_6} [Bornology β] (f : αβ) :

Inverse image of a bornology.

Equations
Instances For
    instance instBornologySubtype {α : Type u_1} [Bornology α] {p : αProp} :
    Equations

    Bounded sets in α × β #

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

    Bounded sets in Π i, π i #

    theorem Bornology.cobounded_pi {ι : Type u_3} {π : ιType u_4} [(i : ι) → Bornology (π i)] :
    Bornology.cobounded ((i : ι) → π i) = Filter.coprodᵢ fun (i : ι) => Bornology.cobounded (π i)
    theorem Bornology.forall_isBounded_image_eval_iff {ι : Type u_3} {π : ιType u_4} [(i : ι) → Bornology (π i)] {s : Set ((i : ι) → π i)} :
    theorem Bornology.IsBounded.image_eval {ι : Type u_3} {π : ιType u_4} [(i : ι) → Bornology (π i)] {s : Set ((i : ι) → π i)} (hs : Bornology.IsBounded s) (i : ι) :
    theorem Bornology.IsBounded.pi {ι : Type u_3} {π : ιType u_4} [(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_3} {π : ιType u_4} [(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_3} {π : ιType u_4} [(i : ι) → Bornology (π i)] {S : (i : ι) → Set (π i)} :
    Bornology.IsBounded (Set.pi Set.univ S) (∃ (i : ι), S i = ) ∀ (i : ι), Bornology.IsBounded (S i)

    Bounded sets in {x // p x} #

    theorem Bornology.isBounded_induced {α : Type u_5} {β : Type u_6} [Bornology β] {f : αβ} {s : Set α} :
    theorem Bornology.isBounded_image_subtype_val {α : Type u_1} [Bornology α] {p : αProp} {s : Set { x : α // p x }} :

    Bounded spaces #

    instance instBoundedSpaceProdInstBornology {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] [BoundedSpace α] [BoundedSpace β] :
    Equations
    • =
    instance instBoundedSpaceForAllInstBornology {ι : Type u_3} {π : ιType u_4} [(i : ι) → Bornology (π i)] [∀ (i : ι), BoundedSpace (π i)] :
    BoundedSpace ((i : ι) → π i)
    Equations
    • =
    theorem boundedSpace_induced_iff {α : Type u_5} {β : Type u_6} [Bornology β] {f : αβ} :
    theorem boundedSpace_subtype_iff {α : Type u_1} [Bornology α] {p : αProp} :
    theorem Bornology.IsBounded.boundedSpace_subtype {α : Type u_1} [Bornology α] {p : αProp} :

    Alias of the reverse direction of boundedSpace_subtype_iff.

    Alias of the reverse direction of boundedSpace_val_set_iff.

    Equations
    • =

    Additive, Multiplicative #

    The bornology on those type synonyms is inherited without change.

    instance instBornologyAdditive {α : Type u_1} [Bornology α] :
    Equations
    • instBornologyAdditive = inst
    Equations
    • instBornologyMultiplicative = inst

    Order dual #

    The bornology on this type synonym is inherited without change.

    Equations
    • instBornologyOrderDual = inst