mathlib documentation

data.set.countable

Countable sets #

def set.countable {α : Type u} (s : set α) :
Prop

A set is countable if there exists an encoding of the set into the natural numbers. An encoding is an injection with a partial inverse, which can be viewed as a constructive analogue of countability. (For the most part, theorems about countable will be classical and encodable will be constructive.)

Equations
theorem set.countable_iff_exists_injective {α : Type u} {s : set α} :
s.countable ∃ (f : s → ), function.injective f
theorem set.countable_iff_exists_inj_on {α : Type u} {s : set α} :
s.countable ∃ (f : α → ), set.inj_on f s

A set s : set α is countable if and only if there exists a function α → ℕ injective on s.

theorem set.countable_iff_exists_surjective {α : Type u} [ne : nonempty α] {s : set α} :
s.countable ∃ (f : → α), s set.range f
theorem set.countable_iff_exists_surjective_to_subtype {α : Type u} {s : set α} (hs : s.nonempty) :

A non-empty set is countable iff there exists a surjection from the natural numbers onto the subtype induced by the set.

def set.countable.to_encodable {α : Type u} {s : set α} :

Convert countable s to encodable s (noncomputable).

Equations
theorem set.countable_encodable' {α : Type u} (s : set α) [H : encodable s] :
theorem set.countable_encodable {α : Type u} [encodable α] (s : set α) :
theorem set.countable.exists_surjective {α : Type u} {s : set α} (hc : s.countable) (hs : s.nonempty) :
∃ (f : → α), s = set.range f

If s : set α is a nonempty countable set, then there exists a map f : ℕ → α such that s = range f.

@[simp]
theorem set.countable_empty {α : Type u} :
@[simp]
theorem set.countable_singleton {α : Type u} (a : α) :
theorem set.countable.mono {α : Type u} {s₁ s₂ : set α} (h : s₁ s₂) :
s₂.countable → s₁.countable
theorem set.countable.image {α : Type u} {β : Type v} {s : set α} (hs : s.countable) (f : α → β) :
theorem set.countable_range {α : Type u} {β : Type v} [encodable α] (f : α → β) :
theorem set.exists_seq_supr_eq_top_iff_countable {α : Type u} [complete_lattice α] {p : α → Prop} (h : ∃ (x : α), p x) :
(∃ (s : → α), (∀ (n : ), p (s n)) (⨆ (n : ), s n) = ) ∃ (S : set α), S.countable (∀ (s : α), s Sp s) Sup S =
theorem set.exists_seq_cover_iff_countable {α : Type u} {p : set α → Prop} (h : ∃ (s : set α), p s) :
(∃ (s : set α), (∀ (n : ), p (s n)) (⋃ (n : ), s n) = set.univ) ∃ (S : set (set α)), S.countable (∀ (s : set α), s Sp s) ⋃₀S = set.univ
theorem set.countable_of_injective_of_countable_image {α : Type u} {β : Type v} {s : set α} {f : α → β} (hf : set.inj_on f s) (hs : (f '' s).countable) :
theorem set.countable_Union {α : Type u} {β : Type v} {t : α → set β} [encodable α] (ht : ∀ (a : α), (t a).countable) :
(⋃ (a : α), t a).countable
theorem set.countable.bUnion {α : Type u} {β : Type v} {s : set α} {t : Π (x : α), x sset β} (hs : s.countable) (ht : ∀ (a : α) (H : a s), (t a H).countable) :
(⋃ (a : α) (H : a s), t a H).countable
theorem set.countable.sUnion {α : Type u} {s : set (set α)} (hs : s.countable) (h : ∀ (a : set α), a s → a.countable) :
theorem set.countable_Union_Prop {β : Type v} {p : Prop} {t : p → set β} (ht : ∀ (h : p), (t h).countable) :
(⋃ (h : p), t h).countable
theorem set.countable.union {α : Type u} {s₁ s₂ : set α} (h₁ : s₁.countable) (h₂ : s₂.countable) :
(s₁ s₂).countable
theorem set.countable.insert {α : Type u} {s : set α} (a : α) (h : s.countable) :
theorem set.finite.countable {α : Type u} {s : set α} :
theorem set.countable_set_of_finite_subset {α : Type u} {s : set α} :
s.countable{t : set α | t.finite t s}.countable

The set of finite subsets of a countable set is countable.

theorem set.countable_pi {α : Type u} {π : α → Type u_1} [fintype α] {s : Π (a : α), set («π» a)} (hs : ∀ (a : α), (s a).countable) :
{f : Π (a : α), «π» a | ∀ (a : α), f a s a}.countable
theorem set.countable.prod {α : Type u} {β : Type v} {s : set α} {t : set β} (hs : s.countable) (ht : t.countable) :
theorem set.countable.image2 {α : Type u} {β : Type v} {γ : Type w} {s : set α} {t : set β} (hs : s.countable) (ht : t.countable) (f : α → β → γ) :
def set.enumerate_countable {α : Type u} {s : set α} (h : s.countable) (default : α) :
→ α

Enumerate elements in a countable set.

Equations
theorem set.subset_range_enumerate {α : Type u} {s : set α} (h : s.countable) (default : α) :
theorem finset.countable_to_set {α : Type u} (s : finset α) :