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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)
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    instance Pi.instBornology {ι : Type u_3} {π : ιType u_4} [(i : ι) → Bornology (π i)] :
    Bornology ((i : ι) → π i)
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      @[reducible, inline]
      abbrev Bornology.induced {α : Type u_5} {β : Type u_6} [Bornology β] (f : αβ) :

      Inverse image of a bornology.

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        Instances For
          instance instBornologySubtype {α : Type u_1} [Bornology α] {p : αProp} :
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            Bounded sets in α × β #

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

            Bounded sets in Π i, π i #

            theorem Bornology.cobounded_pi {ι : Type u_3} {π : ιType u_4} [(i : ι) → Bornology (π i)] :
            cobounded ((i : ι) → π i) = Filter.coprodᵢ fun (i : ι) => 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)} :
            (∀ (i : ι), IsBounded (Function.eval i '' s)) IsBounded s
            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 : 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), 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.univ.pi S).Nonempty) :
            IsBounded (Set.univ.pi S) ∀ (i : ι), 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)} :
            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 : 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 instBoundedSpaceProd {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] [BoundedSpace α] [BoundedSpace β] :
            instance instBoundedSpaceForall {ι : Type u_3} {π : ιType u_4} [(i : ι) → Bornology (π i)] [∀ (i : ι), BoundedSpace (π i)] :
            BoundedSpace ((i : ι) → π i)
            theorem boundedSpace_induced_iff {α : Type u_5} {β : Type u_6} [Bornology β] {f : αβ} :
            theorem Bornology.IsBounded.boundedSpace_subtype {α : Type u_1} [Bornology α] {p : αProp} :

            Alias of the reverse direction of boundedSpace_subtype_iff.

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

            Alias of the reverse direction of boundedSpace_val_set_iff.

            instance instBoundedSpaceSubtype {α : Type u_1} [Bornology α] [BoundedSpace α] {p : αProp} :

            Additive, Multiplicative #

            The bornology on those type synonyms is inherited without change.

            instance instBornologyAdditive {α : Type u_1} [Bornology α] :
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              Order dual #

              The bornology on this type synonym is inherited without change.

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