Filter bases

A filter basis B : FilterBasis α on a type α is a nonempty collection of sets of α such that the intersection of two elements of this collection contains some element of the collection. Compared to filters, filter bases do not require that any set containing an element of B belongs to B. A filter basis B can be used to construct B.filter : Filter α such that a set belongs to B.filter if and only if it contains an element of B.

Given an indexing type ι, a predicate p : ι → Prop, and a map s : ι → Set α, the proposition h : Filter.IsBasis p s makes sure the range of s bounded by p (ie. s '' setOf p) defines a filter basis h.filterBasis.

If one already has a filter l on α, Filter.HasBasis l p s (where p : ι → Prop and s : ι → Set α as above) means that a set belongs to l if and only if it contains some s i with p i. It implies h : Filter.IsBasis p s, and l = h.filterBasis.filter. The point of this definition is that checking statements involving elements of l often reduces to checking them on the basis elements.

We define a function HasBasis.index (h : Filter.HasBasis l p s) (t) (ht : t ∈ l) that returns some index i such that p i and s i ⊆ t. This function can be useful to avoid manual destruction of h.mem_iff.mpr ht using cases or let.

This file also introduces more restricted classes of bases, involving monotonicity or countability. In particular, for l : Filter α, l.IsCountablyGenerated means there is a countable set of sets which generates s. This is reformulated in term of bases, and consequences are derived.

Main statements

• Filter.HasBasis.mem_iff, HasBasis.mem_of_superset, HasBasis.mem_of_mem : restate t ∈ f in terms of a basis;

• Filter.basis_sets : all sets of a filter form a basis;

• Filter.HasBasis.inf, Filter.HasBasis.inf_principal, Filter.HasBasis.prod, Filter.HasBasis.prod_self, Filter.HasBasis.map, Filter.HasBasis.comap : combinators to construct filters of l ⊓ l', l ⊓ 𝓟 t, l ×ˢ l', l ×ˢ l, l.map f, l.comap f respectively;

• Filter.HasBasis.le_iff, Filter.HasBasis.ge_iff, Filter.HasBasis.le_basis_iff : restate l ≤ l' in terms of bases.

• Filter.HasBasis.tendsto_right_iff, Filter.HasBasis.tendsto_left_iff, Filter.HasBasis.tendsto_iff : restate Tendsto f l l' in terms of bases.

• isCountablyGenerated_iff_exists_antitone_basis : proves a filter is countably generated if and only if it admits a basis parametrized by a decreasing sequence of sets indexed by ℕ.

• tendsto_iff_seq_tendsto : an abstract version of "sequentially continuous implies continuous".

Implementation notes

As with Set.iUnion/biUnion/Set.sUnion, there are three different approaches to filter bases:

• Filter.HasBasis l s, s : Set (Set α);
• Filter.HasBasis l s, s : ι → Set α;
• Filter.HasBasis l p s, p : ι → Prop, s : ι → Set α.

We use the latter one because, e.g., 𝓝 x in an EMetricSpace or in a MetricSpace has a basis of this form. The other two can be emulated using s = id or p = fun _ ↦ True.

With this approach sometimes one needs to simp the statement provided by the Filter.HasBasis machinery, e.g., simp only [true_and_iff] or simp only [forall_const] can help with the case p = fun _ ↦ True.

structure FilterBasis (α : Type u_6) : Type u_6

A filter basis B on a type α is a nonempty collection of sets of α such that the intersection of two elements of this collection contains some element of the collection.

• sets : Set (Set α)

  Sets of a filter basis.

• nonempty : self.sets.Nonempty

  The set of filter basis sets is nonempty.

• inter_sets : ∀ {x y : Set α}, x ∈ self.sets → y ∈ self.sets → ∃ z ∈ self.sets, z ⊆ x ∩ y

  The set of filter basis sets is directed downwards.

theorem FilterBasis.nonempty {α : Type u_6} (self : FilterBasis α) : self.sets.Nonempty

The set of filter basis sets is nonempty.

theorem FilterBasis.inter_sets {α : Type u_6} (self : FilterBasis α) {x : Set α} {y : Set α} : x ∈ self.sets → y ∈ self.sets → ∃ z ∈ self.sets, z ⊆ x ∩ y

The set of filter basis sets is directed downwards.

instance FilterBasis.nonempty_sets {α : Type u_1} (B : FilterBasis α) : Nonempty ↑B.sets

Equations

• ⋯ = ⋯

instance instMembershipSetFilterBasis {α : Type u_6} : Membership (Set α) (FilterBasis α)

If B is a filter basis on α, and U a subset of α then we can write U ∈ B as on paper.

Equations

• instMembershipSetFilterBasis = { mem := fun (B : FilterBasis α) (U : Set α) => U ∈ B.sets }

@[simp]

theorem FilterBasis.mem_sets {α : Type u_1} {s : Set α} {B : FilterBasis α} : s ∈ B.sets ↔ s ∈ B

instance instInhabitedFilterBasisNat : Inhabited (FilterBasis ℕ)

Equations

• instInhabitedFilterBasisNat = { default := { sets := Set.range Set.Ici, nonempty := instInhabitedFilterBasisNat.proof_1, inter_sets := @instInhabitedFilterBasisNat.proof_2 } }

def Filter.asBasis {α : Type u_1} (f : Filter α) : FilterBasis α

View a filter as a filter basis.

Equations

• f.asBasis = { sets := f.sets, nonempty := ⋯, inter_sets := ⋯ }

structure Filter.IsBasis {α : Type u_1} {ι : Sort u_4} (p : ι → Prop) (s : ι → Set α) : Prop

IsBasis p s means the image of s bounded by p is a filter basis.

• nonempty : ∃ (i : ι), p i

  There exists at least one i that satisfies p.

• inter : ∀ {i j : ι}, p i → p j → ∃ (k : ι), p k ∧ s k ⊆ s i ∩ s j

  s is directed downwards on i such that p i.

theorem Filter.IsBasis.nonempty {α : Type u_1} {ι : Sort u_4} {p : ι → Prop} {s : ι → Set α} (self : Filter.IsBasis p s) : ∃ (i : ι), p i

There exists at least one i that satisfies p.

theorem Filter.IsBasis.inter {α : Type u_1} {ι : Sort u_4} {p : ι → Prop} {s : ι → Set α} (self : Filter.IsBasis p s) {i : ι} {j : ι} : p i → p j → ∃ (k : ι), p k ∧ s k ⊆ s i ∩ s j

s is directed downwards on i such that p i.

def Filter.IsBasis.filterBasis {α : Type u_1} {ι : Sort u_4} {p : ι → Prop} {s : ι → Set α} (h : Filter.IsBasis p s) : FilterBasis α

Constructs a filter basis from an indexed family of sets satisfying IsBasis.

Equations

• h.filterBasis = { sets := {t : Set α | ∃ (i : ι), p i ∧ s i = t}, nonempty := ⋯, inter_sets := ⋯ }

theorem Filter.IsBasis.mem_filterBasis_iff {α : Type u_1} {ι : Sort u_4} {p : ι → Prop} {s : ι → Set α} (h : Filter.IsBasis p s) {U : Set α} : U ∈ h.filterBasis ↔ ∃ (i : ι), p i ∧ s i = U

def FilterBasis.filter {α : Type u_1} (B : FilterBasis α) : Filter α

The filter associated to a filter basis.

Equations

• B.filter = { sets := {s : Set α | ∃ t ∈ B, t ⊆ s}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ }

theorem FilterBasis.mem_filter_iff {α : Type u_1} (B : FilterBasis α) {U : Set α} : U ∈ B.filter ↔ ∃ s ∈ B, s ⊆ U

theorem FilterBasis.mem_filter_of_mem {α : Type u_1} (B : FilterBasis α) {U : Set α} : U ∈ B → U ∈ B.filter

theorem FilterBasis.eq_iInf_principal {α : Type u_1} (B : FilterBasis α) : B.filter = ⨅ (s : ↑B.sets), Filter.principal ↑s

theorem FilterBasis.generate {α : Type u_1} (B : FilterBasis α) : Filter.generate B.sets = B.filter

def Filter.IsBasis.filter {α : Type u_1} {ι : Sort u_4} {p : ι → Prop} {s : ι → Set α} (h : Filter.IsBasis p s) : Filter α

Constructs a filter from an indexed family of sets satisfying IsBasis.

Equations

• h.filter = h.filterBasis.filter

theorem Filter.IsBasis.mem_filter_iff {α : Type u_1} {ι : Sort u_4} {p : ι → Prop} {s : ι → Set α} (h : Filter.IsBasis p s) {U : Set α} : U ∈ h.filter ↔ ∃ (i : ι), p i ∧ s i ⊆ U

theorem Filter.IsBasis.filter_eq_generate {α : Type u_1} {ι : Sort u_4} {p : ι → Prop} {s : ι → Set α} (h : Filter.IsBasis p s) : h.filter = Filter.generate {U : Set α | ∃ (i : ι), p i ∧ s i = U}

structure Filter.HasBasis {α : Type u_1} {ι : Sort u_4} (l : Filter α) (p : ι → Prop) (s : ι → Set α) : Prop

We say that a filter l has a basis s : ι → Set α bounded by p : ι → Prop, if t ∈ l if and only if t includes s i for some i such that p i.

• mem_iff' : ∀ (t : Set α), t ∈ l ↔ ∃ (i : ι), p i ∧ s i ⊆ t

  A set t belongs to a filter l iff it includes an element of the basis.

theorem Filter.HasBasis.mem_iff' {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (self : l.HasBasis p s) (t : Set α) : t ∈ l ↔ ∃ (i : ι), p i ∧ s i ⊆ t

A set t belongs to a filter l iff it includes an element of the basis.

theorem Filter.hasBasis_generate {α : Type u_1} (s : Set (Set α)) : (Filter.generate s).HasBasis (fun (t : Set (Set α)) => t.Finite ∧ t ⊆ s) fun (t : Set (Set α)) => ⋂₀ t

def Filter.FilterBasis.ofSets {α : Type u_1} (s : Set (Set α)) : FilterBasis α

The smallest filter basis containing a given collection of sets.

Equations

• Filter.FilterBasis.ofSets s = { sets := Set.sInter '' {t : Set (Set α) | t.Finite ∧ t ⊆ s}, nonempty := ⋯, inter_sets := ⋯ }

theorem Filter.FilterBasis.ofSets_sets {α : Type u_1} (s : Set (Set α)) : (Filter.FilterBasis.ofSets s).sets = Set.sInter '' {t : Set (Set α) | t.Finite ∧ t ⊆ s}

theorem Filter.HasBasis.mem_iff {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {t : Set α} (hl : l.HasBasis p s) : t ∈ l ↔ ∃ (i : ι), p i ∧ s i ⊆ t

Definition of HasBasis unfolded with implicit set argument.

theorem Filter.HasBasis.eq_of_same_basis {α : Type u_1} {ι : Sort u_4} {l : Filter α} {l' : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) (hl' : l'.HasBasis p s) : l = l'

theorem Filter.hasBasis_iff {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} : l.HasBasis p s ↔ ∀ (t : Set α), t ∈ l ↔ ∃ (i : ι), p i ∧ s i ⊆ t

theorem Filter.HasBasis.ex_mem {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) : ∃ (i : ι), p i

theorem Filter.HasBasis.nonempty {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) : Nonempty ι

theorem Filter.IsBasis.hasBasis {α : Type u_1} {ι : Sort u_4} {p : ι → Prop} {s : ι → Set α} (h : Filter.IsBasis p s) : h.filter.HasBasis p s

theorem Filter.HasBasis.mem_of_superset {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {t : Set α} {i : ι} (hl : l.HasBasis p s) (hi : p i) (ht : s i ⊆ t) : t ∈ l

theorem Filter.HasBasis.mem_of_mem {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {i : ι} (hl : l.HasBasis p s) (hi : p i) : s i ∈ l

noncomputable def Filter.HasBasis.index {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) (t : Set α) (ht : t ∈ l) : { i : ι // p i }

Index of a basis set such that s i ⊆ t as an element of Subtype p.

Equations

• h.index t ht = ⟨⋯.choose, ⋯⟩

theorem Filter.HasBasis.property_index {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {t : Set α} (h : l.HasBasis p s) (ht : t ∈ l) : p ↑(h.index t ht)

theorem Filter.HasBasis.set_index_mem {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {t : Set α} (h : l.HasBasis p s) (ht : t ∈ l) : s ↑(h.index t ht) ∈ l

theorem Filter.HasBasis.set_index_subset {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {t : Set α} (h : l.HasBasis p s) (ht : t ∈ l) : s ↑(h.index t ht) ⊆ t

theorem Filter.HasBasis.isBasis {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) : Filter.IsBasis p s

theorem Filter.HasBasis.filter_eq {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) : ⋯.filter = l

theorem Filter.HasBasis.eq_generate {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) : l = Filter.generate {U : Set α | ∃ (i : ι), p i ∧ s i = U}

theorem Filter.generate_eq_generate_inter {α : Type u_1} (s : Set (Set α)) : Filter.generate s = Filter.generate (Set.sInter '' {t : Set (Set α) | t.Finite ∧ t ⊆ s})

theorem Filter.ofSets_filter_eq_generate {α : Type u_1} (s : Set (Set α)) : (Filter.FilterBasis.ofSets s).filter = Filter.generate s

theorem FilterBasis.hasBasis {α : Type u_1} (B : FilterBasis α) : B.filter.HasBasis (fun (s : Set α) => s ∈ B) id

theorem Filter.HasBasis.to_hasBasis' {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {p' : ι' → Prop} {s' : ι' → Set α} (hl : l.HasBasis p s) (h : ∀ (i : ι), p i → ∃ (i' : ι'), p' i' ∧ s' i' ⊆ s i) (h' : ∀ (i' : ι'), p' i' → s' i' ∈ l) : l.HasBasis p' s'

theorem Filter.HasBasis.to_hasBasis {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {p' : ι' → Prop} {s' : ι' → Set α} (hl : l.HasBasis p s) (h : ∀ (i : ι), p i → ∃ (i' : ι'), p' i' ∧ s' i' ⊆ s i) (h' : ∀ (i' : ι'), p' i' → ∃ (i : ι), p i ∧ s i ⊆ s' i') : l.HasBasis p' s'

theorem Filter.HasBasis.congr {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) {p' : ι → Prop} {s' : ι → Set α} (hp : ∀ (i : ι), p i ↔ p' i) (hs : ∀ (i : ι), p i → s i = s' i) : l.HasBasis p' s'

theorem Filter.HasBasis.to_subset {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) {t : ι → Set α} (h : ∀ (i : ι), p i → t i ⊆ s i) (ht : ∀ (i : ι), p i → t i ∈ l) : l.HasBasis p t

theorem Filter.HasBasis.eventually_iff {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) {q : α → Prop} : (∀ᶠ (x : α) in l, q x) ↔ ∃ (i : ι), p i ∧ ∀ ⦃x : α⦄, x ∈ s i → q x

theorem Filter.HasBasis.frequently_iff {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) {q : α → Prop} : (∃ᶠ (x : α) in l, q x) ↔ ∀ (i : ι), p i → ∃ x ∈ s i, q x

theorem Filter.HasBasis.exists_iff {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) {P : Set α → Prop} (mono : ∀ ⦃s t : Set α⦄, s ⊆ t → P t → P s) : (∃ s ∈ l, P s) ↔ ∃ (i : ι), p i ∧ P (s i)

theorem Filter.HasBasis.forall_iff {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) {P : Set α → Prop} (mono : ∀ ⦃s t : Set α⦄, s ⊆ t → P s → P t) : (∀ s ∈ l, P s) ↔ ∀ (i : ι), p i → P (s i)

theorem Filter.HasBasis.neBot_iff {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) : l.NeBot ↔ ∀ {i : ι}, p i → (s i).Nonempty

theorem Filter.HasBasis.eq_bot_iff {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) : l = ⊥ ↔ ∃ (i : ι), p i ∧ s i = ∅

theorem Filter.generate_neBot_iff {α : Type u_1} {s : Set (Set α)} : (Filter.generate s).NeBot ↔ ∀ t ⊆ s, t.Finite → (⋂₀ t).Nonempty

theorem Filter.basis_sets {α : Type u_1} (l : Filter α) : l.HasBasis (fun (s : Set α) => s ∈ l) id

theorem Filter.asBasis_filter {α : Type u_1} (f : Filter α) : f.asBasis.filter = f

theorem Filter.hasBasis_self {α : Type u_1} {l : Filter α} {P : Set α → Prop} : l.HasBasis (fun (s : Set α) => s ∈ l ∧ P s) id ↔ ∀ t ∈ l, ∃ r ∈ l, P r ∧ r ⊆ t

theorem Filter.HasBasis.comp_surjective {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) {g : ι' → ι} (hg : Function.Surjective g) : l.HasBasis (p ∘ g) (s ∘ g)

theorem Filter.HasBasis.comp_equiv {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) (e : ι' ≃ ι) : l.HasBasis (p ∘ ⇑e) (s ∘ ⇑e)

theorem Filter.HasBasis.to_image_id' {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) : l.HasBasis (fun (t : Set α) => ∃ (i : ι), p i ∧ s i = t) id

theorem Filter.HasBasis.to_image_id {α : Type u_1} {l : Filter α} {ι : Type u_6} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) : l.HasBasis (fun (x : Set α) => x ∈ s '' {i : ι | p i}) id

theorem Filter.HasBasis.restrict {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) {q : ι → Prop} (hq : ∀ (i : ι), p i → ∃ (j : ι), p j ∧ q j ∧ s j ⊆ s i) : l.HasBasis (fun (i : ι) => p i ∧ q i) s

If {s i | p i} is a basis of a filter l and each s i includes s j such that p j ∧ q j, then {s j | p j ∧ q j} is a basis of l.

theorem Filter.HasBasis.restrict_subset {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) {V : Set α} (hV : V ∈ l) : l.HasBasis (fun (i : ι) => p i ∧ s i ⊆ V) s

If {s i | p i} is a basis of a filter l and V ∈ l, then {s i | p i ∧ s i ⊆ V} is a basis of l.

theorem Filter.HasBasis.hasBasis_self_subset {α : Type u_1} {l : Filter α} {p : Set α → Prop} (h : l.HasBasis (fun (s : Set α) => s ∈ l ∧ p s) id) {V : Set α} (hV : V ∈ l) : l.HasBasis (fun (s : Set α) => s ∈ l ∧ p s ∧ s ⊆ V) id

theorem Filter.HasBasis.ge_iff {α : Type u_1} {ι' : Sort u_5} {l : Filter α} {l' : Filter α} {p' : ι' → Prop} {s' : ι' → Set α} (hl' : l'.HasBasis p' s') : l ≤ l' ↔ ∀ (i' : ι'), p' i' → s' i' ∈ l

theorem Filter.HasBasis.le_iff {α : Type u_1} {ι : Sort u_4} {l : Filter α} {l' : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) : l ≤ l' ↔ ∀ t ∈ l', ∃ (i : ι), p i ∧ s i ⊆ t

theorem Filter.HasBasis.le_basis_iff {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l : Filter α} {l' : Filter α} {p : ι → Prop} {s : ι → Set α} {p' : ι' → Prop} {s' : ι' → Set α} (hl : l.HasBasis p s) (hl' : l'.HasBasis p' s') : l ≤ l' ↔ ∀ (i' : ι'), p' i' → ∃ (i : ι), p i ∧ s i ⊆ s' i'

theorem Filter.HasBasis.ext {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l : Filter α} {l' : Filter α} {p : ι → Prop} {s : ι → Set α} {p' : ι' → Prop} {s' : ι' → Set α} (hl : l.HasBasis p s) (hl' : l'.HasBasis p' s') (h : ∀ (i : ι), p i → ∃ (i' : ι'), p' i' ∧ s' i' ⊆ s i) (h' : ∀ (i' : ι'), p' i' → ∃ (i : ι), p i ∧ s i ⊆ s' i') : l = l'

theorem Filter.HasBasis.inf' {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l : Filter α} {l' : Filter α} {p : ι → Prop} {s : ι → Set α} {p' : ι' → Prop} {s' : ι' → Set α} (hl : l.HasBasis p s) (hl' : l'.HasBasis p' s') : (l ⊓ l').HasBasis (fun (i : ι ×' ι') => p i.fst ∧ p' i.snd) fun (i : ι ×' ι') => s i.fst ∩ s' i.snd

theorem Filter.HasBasis.inf {α : Type u_1} {l : Filter α} {l' : Filter α} {ι : Type u_6} {ι' : Type u_7} {p : ι → Prop} {s : ι → Set α} {p' : ι' → Prop} {s' : ι' → Set α} (hl : l.HasBasis p s) (hl' : l'.HasBasis p' s') : (l ⊓ l').HasBasis (fun (i : ι × ι') => p i.1 ∧ p' i.2) fun (i : ι × ι') => s i.1 ∩ s' i.2

theorem Filter.hasBasis_iInf' {α : Type u_1} {ι : Type u_6} {ι' : ι → Type u_7} {l : ι → Filter α} {p : (i : ι) → ι' i → Prop} {s : (i : ι) → ι' i → Set α} (hl : ∀ (i : ι), (l i).HasBasis (p i) (s i)) : (⨅ (i : ι), l i).HasBasis (fun (If : Set ι × ((i : ι) → ι' i)) => If.1.Finite ∧ ∀ i ∈ If.1, p i (If.2 i)) fun (If : Set ι × ((i : ι) → ι' i)) => ⋂ i ∈ If.1, s i (If.2 i)

theorem Filter.hasBasis_iInf {α : Type u_1} {ι : Type u_6} {ι' : ι → Type u_7} {l : ι → Filter α} {p : (i : ι) → ι' i → Prop} {s : (i : ι) → ι' i → Set α} (hl : ∀ (i : ι), (l i).HasBasis (p i) (s i)) : (⨅ (i : ι), l i).HasBasis (fun (If : (I : Set ι) × ((i : ↑I) → ι' ↑i)) => If.fst.Finite ∧ ∀ (i : ↑If.fst), p (↑i) (If.snd i)) fun (If : (I : Set ι) × ((i : ↑I) → ι' ↑i)) => ⋂ (i : ↑If.fst), s (↑i) (If.snd i)

theorem Filter.hasBasis_iInf_of_directed' {α : Type u_1} {ι : Type u_6} {ι' : ι → Type u_7} [Nonempty ι] {l : ι → Filter α} (s : (i : ι) → ι' i → Set α) (p : (i : ι) → ι' i → Prop) (hl : ∀ (i : ι), (l i).HasBasis (p i) (s i)) (h : Directed (fun (x1 x2 : Filter α) => x1 ≥ x2) l) : (⨅ (i : ι), l i).HasBasis (fun (ii' : (i : ι) × ι' i) => p ii'.fst ii'.snd) fun (ii' : (i : ι) × ι' i) => s ii'.fst ii'.snd

theorem Filter.hasBasis_iInf_of_directed {α : Type u_1} {ι : Type u_6} {ι' : Type u_7} [Nonempty ι] {l : ι → Filter α} (s : ι → ι' → Set α) (p : ι → ι' → Prop) (hl : ∀ (i : ι), (l i).HasBasis (p i) (s i)) (h : Directed (fun (x1 x2 : Filter α) => x1 ≥ x2) l) : (⨅ (i : ι), l i).HasBasis (fun (ii' : ι × ι') => p ii'.1 ii'.2) fun (ii' : ι × ι') => s ii'.1 ii'.2

theorem Filter.hasBasis_biInf_of_directed' {α : Type u_1} {ι : Type u_6} {ι' : ι → Type u_7} {dom : Set ι} (hdom : dom.Nonempty) {l : ι → Filter α} (s : (i : ι) → ι' i → Set α) (p : (i : ι) → ι' i → Prop) (hl : ∀ i ∈ dom, (l i).HasBasis (p i) (s i)) (h : DirectedOn (l ⁻¹'o GE.ge) dom) : (⨅ i ∈ dom, l i).HasBasis (fun (ii' : (i : ι) × ι' i) => ii'.fst ∈ dom ∧ p ii'.fst ii'.snd) fun (ii' : (i : ι) × ι' i) => s ii'.fst ii'.snd

theorem Filter.hasBasis_biInf_of_directed {α : Type u_1} {ι : Type u_6} {ι' : Type u_7} {dom : Set ι} (hdom : dom.Nonempty) {l : ι → Filter α} (s : ι → ι' → Set α) (p : ι → ι' → Prop) (hl : ∀ i ∈ dom, (l i).HasBasis (p i) (s i)) (h : DirectedOn (l ⁻¹'o GE.ge) dom) : (⨅ i ∈ dom, l i).HasBasis (fun (ii' : ι × ι') => ii'.1 ∈ dom ∧ p ii'.1 ii'.2) fun (ii' : ι × ι') => s ii'.1 ii'.2

theorem Filter.hasBasis_principal {α : Type u_1} (t : Set α) : (Filter.principal t).HasBasis (fun (x : Unit) => True) fun (x : Unit) => t

theorem Filter.hasBasis_pure {α : Type u_1} (x : α) : (pure x).HasBasis (fun (x : Unit) => True) fun (x_1 : Unit) => {x}

theorem Filter.HasBasis.sup' {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l : Filter α} {l' : Filter α} {p : ι → Prop} {s : ι → Set α} {p' : ι' → Prop} {s' : ι' → Set α} (hl : l.HasBasis p s) (hl' : l'.HasBasis p' s') : (l ⊔ l').HasBasis (fun (i : ι ×' ι') => p i.fst ∧ p' i.snd) fun (i : ι ×' ι') => s i.fst ∪ s' i.snd

theorem Filter.HasBasis.sup {α : Type u_1} {l : Filter α} {l' : Filter α} {ι : Type u_6} {ι' : Type u_7} {p : ι → Prop} {s : ι → Set α} {p' : ι' → Prop} {s' : ι' → Set α} (hl : l.HasBasis p s) (hl' : l'.HasBasis p' s') : (l ⊔ l').HasBasis (fun (i : ι × ι') => p i.1 ∧ p' i.2) fun (i : ι × ι') => s i.1 ∪ s' i.2

theorem Filter.hasBasis_iSup {α : Type u_1} {ι : Sort u_6} {ι' : ι → Type u_7} {l : ι → Filter α} {p : (i : ι) → ι' i → Prop} {s : (i : ι) → ι' i → Set α} (hl : ∀ (i : ι), (l i).HasBasis (p i) (s i)) : (⨆ (i : ι), l i).HasBasis (fun (f : (i : ι) → ι' i) => ∀ (i : ι), p i (f i)) fun (f : (i : ι) → ι' i) => ⋃ (i : ι), s i (f i)

theorem Filter.HasBasis.sup_principal {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) (t : Set α) : (l ⊔ Filter.principal t).HasBasis p fun (i : ι) => s i ∪ t

theorem Filter.HasBasis.sup_pure {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) (x : α) : (l ⊔ pure x).HasBasis p fun (i : ι) => s i ∪ {x}

theorem Filter.HasBasis.inf_principal {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) (s' : Set α) : (l ⊓ Filter.principal s').HasBasis p fun (i : ι) => s i ∩ s'

theorem Filter.HasBasis.principal_inf {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) (s' : Set α) : (Filter.principal s' ⊓ l).HasBasis p fun (i : ι) => s' ∩ s i

theorem Filter.HasBasis.inf_basis_neBot_iff {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l : Filter α} {l' : Filter α} {p : ι → Prop} {s : ι → Set α} {p' : ι' → Prop} {s' : ι' → Set α} (hl : l.HasBasis p s) (hl' : l'.HasBasis p' s') : (l ⊓ l').NeBot ↔ ∀ ⦃i : ι⦄, p i → ∀ ⦃i' : ι'⦄, p' i' → (s i ∩ s' i').Nonempty

theorem Filter.HasBasis.inf_neBot_iff {α : Type u_1} {ι : Sort u_4} {l : Filter α} {l' : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) : (l ⊓ l').NeBot ↔ ∀ ⦃i : ι⦄, p i → ∀ ⦃s' : Set α⦄, s' ∈ l' → (s i ∩ s').Nonempty

theorem Filter.HasBasis.inf_principal_neBot_iff {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) {t : Set α} : (l ⊓ Filter.principal t).NeBot ↔ ∀ ⦃i : ι⦄, p i → (s i ∩ t).Nonempty

theorem Filter.HasBasis.disjoint_iff {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l : Filter α} {l' : Filter α} {p : ι → Prop} {s : ι → Set α} {p' : ι' → Prop} {s' : ι' → Set α} (hl : l.HasBasis p s) (hl' : l'.HasBasis p' s') : Disjoint l l' ↔ ∃ (i : ι), p i ∧ ∃ (i' : ι'), p' i' ∧ Disjoint (s i) (s' i')

theorem Disjoint.exists_mem_filter_basis {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l : Filter α} {l' : Filter α} {p : ι → Prop} {s : ι → Set α} {p' : ι' → Prop} {s' : ι' → Set α} (h : Disjoint l l') (hl : l.HasBasis p s) (hl' : l'.HasBasis p' s') : ∃ (i : ι), p i ∧ ∃ (i' : ι'), p' i' ∧ Disjoint (s i) (s' i')

theorem Pairwise.exists_mem_filter_basis_of_disjoint {α : Type u_1} {I : Type u_7} [Finite I] {l : I → Filter α} {ι : I → Sort u_6} {p : (i : I) → ι i → Prop} {s : (i : I) → ι i → Set α} (hd : Pairwise (Disjoint on l)) (h : ∀ (i : I), (l i).HasBasis (p i) (s i)) : ∃ (ind : (i : I) → ι i), (∀ (i : I), p i (ind i)) ∧ Pairwise (Disjoint on fun (i : I) => s i (ind i))

theorem Set.PairwiseDisjoint.exists_mem_filter_basis {α : Type u_1} {I : Type u_6} {l : I → Filter α} {ι : I → Sort u_7} {p : (i : I) → ι i → Prop} {s : (i : I) → ι i → Set α} {S : Set I} (hd : S.PairwiseDisjoint l) (hS : S.Finite) (h : ∀ (i : I), (l i).HasBasis (p i) (s i)) : ∃ (ind : (i : I) → ι i), (∀ (i : I), p i (ind i)) ∧ S.PairwiseDisjoint fun (i : I) => s i (ind i)

theorem Filter.inf_neBot_iff {α : Type u_1} {l : Filter α} {l' : Filter α} : (l ⊓ l').NeBot ↔ ∀ ⦃s : Set α⦄, s ∈ l → ∀ ⦃s' : Set α⦄, s' ∈ l' → (s ∩ s').Nonempty

theorem Filter.inf_principal_neBot_iff {α : Type u_1} {l : Filter α} {s : Set α} : (l ⊓ Filter.principal s).NeBot ↔ ∀ U ∈ l, (U ∩ s).Nonempty

theorem Filter.mem_iff_inf_principal_compl {α : Type u_1} {f : Filter α} {s : Set α} : s ∈ f ↔ f ⊓ Filter.principal sᶜ = ⊥

theorem Filter.not_mem_iff_inf_principal_compl {α : Type u_1} {f : Filter α} {s : Set α} : s ∉ f ↔ (f ⊓ Filter.principal sᶜ).NeBot

@[simp]

theorem Filter.disjoint_principal_right {α : Type u_1} {f : Filter α} {s : Set α} : Disjoint f (Filter.principal s) ↔ sᶜ ∈ f

@[simp]

theorem Filter.disjoint_principal_left {α : Type u_1} {f : Filter α} {s : Set α} : Disjoint (Filter.principal s) f ↔ sᶜ ∈ f

@[simp]

theorem Filter.disjoint_principal_principal {α : Type u_1} {s : Set α} {t : Set α} : Disjoint (Filter.principal s) (Filter.principal t) ↔ Disjoint s t

theorem Disjoint.filter_principal {α : Type u_1} {s : Set α} {t : Set α} : Disjoint s t → Disjoint (Filter.principal s) (Filter.principal t)

Alias of the reverse direction of Filter.disjoint_principal_principal.

@[simp]

theorem Filter.disjoint_pure_pure {α : Type u_1} {x : α} {y : α} : Disjoint (pure x) (pure y) ↔ x ≠ y

theorem Filter.HasBasis.disjoint_iff_left {α : Type u_1} {ι : Sort u_4} {l : Filter α} {l' : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) : Disjoint l l' ↔ ∃ (i : ι), p i ∧ (s i)ᶜ ∈ l'

theorem Filter.HasBasis.disjoint_iff_right {α : Type u_1} {ι : Sort u_4} {l : Filter α} {l' : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) : Disjoint l' l ↔ ∃ (i : ι), p i ∧ (s i)ᶜ ∈ l'

theorem Filter.le_iff_forall_inf_principal_compl {α : Type u_1} {f : Filter α} {g : Filter α} : f ≤ g ↔ ∀ V ∈ g, f ⊓ Filter.principal Vᶜ = ⊥

theorem Filter.inf_neBot_iff_frequently_left {α : Type u_1} {f : Filter α} {g : Filter α} : (f ⊓ g).NeBot ↔ ∀ {p : α → Prop}, (∀ᶠ (x : α) in f, p x) → ∃ᶠ (x : α) in g, p x

theorem Filter.inf_neBot_iff_frequently_right {α : Type u_1} {f : Filter α} {g : Filter α} : (f ⊓ g).NeBot ↔ ∀ {p : α → Prop}, (∀ᶠ (x : α) in g, p x) → ∃ᶠ (x : α) in f, p x

theorem Filter.HasBasis.eq_biInf {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) : l = ⨅ (i : ι), ⨅ (_ : p i), Filter.principal (s i)

theorem Filter.HasBasis.eq_iInf {α : Type u_1} {ι : Sort u_4} {l : Filter α} {s : ι → Set α} (h : l.HasBasis (fun (x : ι) => True) s) : l = ⨅ (i : ι), Filter.principal (s i)

theorem Filter.hasBasis_iInf_principal {α : Type u_1} {ι : Sort u_4} {s : ι → Set α} (h : Directed (fun (x1 x2 : Set α) => x1 ≥ x2) s) [Nonempty ι] : (⨅ (i : ι), Filter.principal (s i)).HasBasis (fun (x : ι) => True) s

theorem Filter.hasBasis_iInf_principal_finite {α : Type u_1} {ι : Type u_6} (s : ι → Set α) : (⨅ (i : ι), Filter.principal (s i)).HasBasis (fun (t : Set ι) => t.Finite) fun (t : Set ι) => ⋂ i ∈ t, s i

If s : ι → Set α is an indexed family of sets, then finite intersections of s i form a basis of ⨅ i, 𝓟 (s i).

theorem Filter.hasBasis_biInf_principal {α : Type u_1} {β : Type u_2} {s : β → Set α} {S : Set β} (h : DirectedOn (s ⁻¹'o fun (x1 x2 : Set α) => x1 ≥ x2) S) (ne : S.Nonempty) : (⨅ i ∈ S, Filter.principal (s i)).HasBasis (fun (i : β) => i ∈ S) s

theorem Filter.hasBasis_biInf_principal' {α : Type u_1} {ι : Type u_6} {p : ι → Prop} {s : ι → Set α} (h : ∀ (i : ι), p i → ∀ (j : ι), p j → ∃ (k : ι), p k ∧ s k ⊆ s i ∧ s k ⊆ s j) (ne : ∃ (i : ι), p i) : (⨅ (i : ι), ⨅ (_ : p i), Filter.principal (s i)).HasBasis p s

theorem Filter.HasBasis.map {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (f : α → β) (hl : l.HasBasis p s) : (Filter.map f l).HasBasis p fun (i : ι) => f '' s i

theorem Filter.HasBasis.comap {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (f : β → α) (hl : l.HasBasis p s) : (Filter.comap f l).HasBasis p fun (i : ι) => f ⁻¹' s i

theorem Filter.comap_hasBasis {α : Type u_1} {β : Type u_2} (f : α → β) (l : Filter β) : (Filter.comap f l).HasBasis (fun (s : Set β) => s ∈ l) fun (s : Set β) => f ⁻¹' s

theorem Filter.HasBasis.forall_mem_mem {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) {x : α} : (∀ t ∈ l, x ∈ t) ↔ ∀ (i : ι), p i → x ∈ s i

theorem Filter.HasBasis.biInf_mem {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} [CompleteLattice β] {f : Set α → β} (h : l.HasBasis p s) (hf : Monotone f) : ⨅ t ∈ l, f t = ⨅ (i : ι), ⨅ (_ : p i), f (s i)

theorem Filter.HasBasis.biInter_mem {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {f : Set α → Set β} (h : l.HasBasis p s) (hf : Monotone f) : ⋂ t ∈ l, f t = ⋂ (i : ι), ⋂ (_ : p i), f (s i)

theorem Filter.HasBasis.ker {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) : l.ker = ⋂ (i : ι), ⋂ (_ : p i), s i

structure Filter.IsAntitoneBasis {α : Type u_1} {ι'' : Type u_6} [Preorder ι''] (s'' : ι'' → Set α) extends Filter.IsBasis : Prop

IsAntitoneBasis s means the image of s is a filter basis such that s is decreasing.

• nonempty : ∃ (i : ι''), True
• inter : ∀ {i j : ι''}, True → True → ∃ (k : ι''), True ∧ s'' k ⊆ s'' i ∩ s'' j
• antitone : Antitone s''

  The sequence of sets is antitone.

theorem Filter.IsAntitoneBasis.antitone {α : Type u_1} {ι'' : Type u_6} [Preorder ι''] {s'' : ι'' → Set α} (self : Filter.IsAntitoneBasis s'') : Antitone s''

The sequence of sets is antitone.

structure Filter.HasAntitoneBasis {α : Type u_1} {ι'' : Type u_6} [Preorder ι''] (l : Filter α) (s : ι'' → Set α) extends Filter.HasBasis : Prop

We say that a filter l has an antitone basis s : ι → Set α, if t ∈ l if and only if t includes s i for some i, and s is decreasing.

• mem_iff' : ∀ (t : Set α), t ∈ l ↔ ∃ (i : ι''), True ∧ s i ⊆ t
• antitone : Antitone s

  The sequence of sets is antitone.

theorem Filter.HasAntitoneBasis.antitone {α : Type u_1} {ι'' : Type u_6} [Preorder ι''] {l : Filter α} {s : ι'' → Set α} (self : l.HasAntitoneBasis s) : Antitone s

The sequence of sets is antitone.

theorem Filter.HasAntitoneBasis.map {α : Type u_1} {β : Type u_2} {ι'' : Type u_6} [Preorder ι''] {l : Filter α} {s : ι'' → Set α} (hf : l.HasAntitoneBasis s) (m : α → β) : (Filter.map m l).HasAntitoneBasis fun (x : ι'') => m '' s x

theorem Filter.HasAntitoneBasis.comap {α : Type u_1} {β : Type u_2} {ι'' : Type u_6} [Preorder ι''] {l : Filter α} {s : ι'' → Set α} (hf : l.HasAntitoneBasis s) (m : β → α) : (Filter.comap m l).HasAntitoneBasis fun (x : ι'') => m ⁻¹' s x

theorem Filter.HasAntitoneBasis.iInf_principal {α : Type u_1} {ι : Type u_7} [Preorder ι] [Nonempty ι] [IsDirected ι fun (x1 x2 : ι) => x1 ≤ x2] {s : ι → Set α} (hs : Antitone s) : (⨅ (i : ι), Filter.principal (s i)).HasAntitoneBasis s

theorem Filter.HasBasis.tendsto_left_iff {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {la : Filter α} {pa : ι → Prop} {sa : ι → Set α} {lb : Filter β} {f : α → β} (hla : la.HasBasis pa sa) : Filter.Tendsto f la lb ↔ ∀ t ∈ lb, ∃ (i : ι), pa i ∧ Set.MapsTo f (sa i) t

theorem Filter.HasBasis.tendsto_right_iff {α : Type u_1} {β : Type u_2} {ι' : Sort u_5} {la : Filter α} {lb : Filter β} {pb : ι' → Prop} {sb : ι' → Set β} {f : α → β} (hlb : lb.HasBasis pb sb) : Filter.Tendsto f la lb ↔ ∀ (i : ι'), pb i → ∀ᶠ (x : α) in la, f x ∈ sb i

theorem Filter.HasBasis.tendsto_iff {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {ι' : Sort u_5} {la : Filter α} {pa : ι → Prop} {sa : ι → Set α} {lb : Filter β} {pb : ι' → Prop} {sb : ι' → Set β} {f : α → β} (hla : la.HasBasis pa sa) (hlb : lb.HasBasis pb sb) : Filter.Tendsto f la lb ↔ ∀ (ib : ι'), pb ib → ∃ (ia : ι), pa ia ∧ ∀ x ∈ sa ia, f x ∈ sb ib

theorem Filter.Tendsto.basis_left {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {la : Filter α} {pa : ι → Prop} {sa : ι → Set α} {lb : Filter β} {f : α → β} (H : Filter.Tendsto f la lb) (hla : la.HasBasis pa sa) (t : Set β) : t ∈ lb → ∃ (i : ι), pa i ∧ Set.MapsTo f (sa i) t

theorem Filter.Tendsto.basis_right {α : Type u_1} {β : Type u_2} {ι' : Sort u_5} {la : Filter α} {lb : Filter β} {pb : ι' → Prop} {sb : ι' → Set β} {f : α → β} (H : Filter.Tendsto f la lb) (hlb : lb.HasBasis pb sb) (i : ι') : pb i → ∀ᶠ (x : α) in la, f x ∈ sb i

theorem Filter.Tendsto.basis_both {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {ι' : Sort u_5} {la : Filter α} {pa : ι → Prop} {sa : ι → Set α} {lb : Filter β} {pb : ι' → Prop} {sb : ι' → Set β} {f : α → β} (H : Filter.Tendsto f la lb) (hla : la.HasBasis pa sa) (hlb : lb.HasBasis pb sb) (ib : ι') : pb ib → ∃ (ia : ι), pa ia ∧ Set.MapsTo f (sa ia) (sb ib)

theorem Filter.HasBasis.prod_pprod {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {ι' : Sort u_5} {la : Filter α} {pa : ι → Prop} {sa : ι → Set α} {lb : Filter β} {pb : ι' → Prop} {sb : ι' → Set β} (hla : la.HasBasis pa sa) (hlb : lb.HasBasis pb sb) : (la ×ˢ lb).HasBasis (fun (i : ι ×' ι') => pa i.fst ∧ pb i.snd) fun (i : ι ×' ι') => sa i.fst ×ˢ sb i.snd

theorem Filter.HasBasis.prod {α : Type u_1} {β : Type u_2} {la : Filter α} {lb : Filter β} {ι : Type u_6} {ι' : Type u_7} {pa : ι → Prop} {sa : ι → Set α} {pb : ι' → Prop} {sb : ι' → Set β} (hla : la.HasBasis pa sa) (hlb : lb.HasBasis pb sb) : (la ×ˢ lb).HasBasis (fun (i : ι × ι') => pa i.1 ∧ pb i.2) fun (i : ι × ι') => sa i.1 ×ˢ sb i.2

theorem Filter.HasBasis.prod_same_index {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {la : Filter α} {sa : ι → Set α} {lb : Filter β} {p : ι → Prop} {sb : ι → Set β} (hla : la.HasBasis p sa) (hlb : lb.HasBasis p sb) (h_dir : ∀ {i j : ι}, p i → p j → ∃ (k : ι), p k ∧ sa k ⊆ sa i ∧ sb k ⊆ sb j) : (la ×ˢ lb).HasBasis p fun (i : ι) => sa i ×ˢ sb i

theorem Filter.HasBasis.prod_same_index_mono {α : Type u_1} {β : Type u_2} {la : Filter α} {lb : Filter β} {ι : Type u_6} [LinearOrder ι] {p : ι → Prop} {sa : ι → Set α} {sb : ι → Set β} (hla : la.HasBasis p sa) (hlb : lb.HasBasis p sb) (hsa : MonotoneOn sa {i : ι | p i}) (hsb : MonotoneOn sb {i : ι | p i}) : (la ×ˢ lb).HasBasis p fun (i : ι) => sa i ×ˢ sb i

theorem Filter.HasBasis.prod_same_index_anti {α : Type u_1} {β : Type u_2} {la : Filter α} {lb : Filter β} {ι : Type u_6} [LinearOrder ι] {p : ι → Prop} {sa : ι → Set α} {sb : ι → Set β} (hla : la.HasBasis p sa) (hlb : lb.HasBasis p sb) (hsa : AntitoneOn sa {i : ι | p i}) (hsb : AntitoneOn sb {i : ι | p i}) : (la ×ˢ lb).HasBasis p fun (i : ι) => sa i ×ˢ sb i

theorem Filter.HasBasis.prod_self {α : Type u_1} {ι : Sort u_4} {la : Filter α} {pa : ι → Prop} {sa : ι → Set α} (hl : la.HasBasis pa sa) : (la ×ˢ la).HasBasis pa fun (i : ι) => sa i ×ˢ sa i

theorem Filter.mem_prod_self_iff {α : Type u_1} {la : Filter α} {s : Set (α × α)} : s ∈ la ×ˢ la ↔ ∃ t ∈ la, t ×ˢ t ⊆ s

theorem Filter.eventually_prod_self_iff {α : Type u_1} {la : Filter α} {r : α → α → Prop} : (∀ᶠ (x : α × α) in la ×ˢ la, r x.1 x.2) ↔ ∃ t ∈ la, ∀ x ∈ t, ∀ y ∈ t, r x y

theorem Filter.HasAntitoneBasis.prod {α : Type u_1} {β : Type u_2} {ι : Type u_6} [LinearOrder ι] {f : Filter α} {g : Filter β} {s : ι → Set α} {t : ι → Set β} (hf : f.HasAntitoneBasis s) (hg : g.HasAntitoneBasis t) : (f ×ˢ g).HasAntitoneBasis fun (n : ι) => s n ×ˢ t n

theorem Filter.HasBasis.coprod {α : Type u_1} {β : Type u_2} {la : Filter α} {lb : Filter β} {ι : Type u_6} {ι' : Type u_7} {pa : ι → Prop} {sa : ι → Set α} {pb : ι' → Prop} {sb : ι' → Set β} (hla : la.HasBasis pa sa) (hlb : lb.HasBasis pb sb) : (la.coprod lb).HasBasis (fun (i : ι × ι') => pa i.1 ∧ pb i.2) fun (i : ι × ι') => Prod.fst ⁻¹' sa i.1 ∪ Prod.snd ⁻¹' sb i.2

theorem Filter.map_sigma_mk_comap {α : Type u_1} {β : Type u_2} {π : α → Type u_6} {π' : β → Type u_7} {f : α → β} (hf : Function.Injective f) (g : (a : α) → π a → π' (f a)) (a : α) (l : Filter (π' (f a))) : Filter.map (Sigma.mk a) (Filter.comap (g a) l) = Filter.comap (Sigma.map f g) (Filter.map (Sigma.mk (f a)) l)

class Filter.IsCountablyGenerated {α : Type u_1} (f : Filter α) : Prop

IsCountablyGenerated f means f = generate s for some countable s.

• out : ∃ (s : Set (Set α)), s.Countable ∧ f = Filter.generate s

  There exists a countable set that generates the filter.

theorem Filter.IsCountablyGenerated.out {α : Type u_1} {f : Filter α} [self : f.IsCountablyGenerated] : ∃ (s : Set (Set α)), s.Countable ∧ f = Filter.generate s

There exists a countable set that generates the filter.

structure Filter.IsCountableBasis {α : Type u_1} {ι : Type u_4} (p : ι → Prop) (s : ι → Set α) extends Filter.IsBasis : Prop

IsCountableBasis p s means the image of s bounded by p is a countable filter basis.

• nonempty : ∃ (i : ι), p i
• inter : ∀ {i j : ι}, p i → p j → ∃ (k : ι), p k ∧ s k ⊆ s i ∩ s j
• countable : (setOf p).Countable

  The set of i that satisfy the predicate p is countable.

theorem Filter.IsCountableBasis.countable {α : Type u_1} {ι : Type u_4} {p : ι → Prop} {s : ι → Set α} (self : Filter.IsCountableBasis p s) : (setOf p).Countable

The set of i that satisfy the predicate p is countable.

structure Filter.HasCountableBasis {α : Type u_1} {ι : Type u_4} (l : Filter α) (p : ι → Prop) (s : ι → Set α) extends Filter.HasBasis : Prop

We say that a filter l has a countable basis s : ι → Set α bounded by p : ι → Prop, if t ∈ l if and only if t includes s i for some i such that p i, and the set defined by p is countable.

• mem_iff' : ∀ (t : Set α), t ∈ l ↔ ∃ (i : ι), p i ∧ s i ⊆ t
• countable : (setOf p).Countable

  The set of i that satisfy the predicate p is countable.

theorem Filter.HasCountableBasis.countable {α : Type u_1} {ι : Type u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (self : l.HasCountableBasis p s) : (setOf p).Countable

The set of i that satisfy the predicate p is countable.

structure Filter.CountableFilterBasis (α : Type u_6) extends FilterBasis : Type u_6

A countable filter basis B on a type α is a nonempty countable collection of sets of α such that the intersection of two elements of this collection contains some element of the collection.

• sets : Set (Set α)
• nonempty : self.sets.Nonempty
• inter_sets : ∀ {x y : Set α}, x ∈ self.sets → y ∈ self.sets → ∃ z ∈ self.sets, z ⊆ x ∩ y
• countable : self.sets.Countable

  The set of sets of the filter basis is countable.

theorem Filter.CountableFilterBasis.countable {α : Type u_6} (self : Filter.CountableFilterBasis α) : self.sets.Countable

The set of sets of the filter basis is countable.

instance Filter.Nat.inhabitedCountableFilterBasis : Inhabited (Filter.CountableFilterBasis ℕ)

Equations

• Filter.Nat.inhabitedCountableFilterBasis = { default := { toFilterBasis := default, countable := Filter.Nat.inhabitedCountableFilterBasis.proof_1 } }

theorem Filter.HasCountableBasis.isCountablyGenerated {α : Type u_1} {ι : Type u_4} {f : Filter α} {p : ι → Prop} {s : ι → Set α} (h : f.HasCountableBasis p s) : f.IsCountablyGenerated

theorem Filter.HasBasis.isCountablyGenerated {α : Type u_1} {ι : Type u_4} [Countable ι] {f : Filter α} {p : ι → Prop} {s : ι → Set α} (h : f.HasBasis p s) : f.IsCountablyGenerated

theorem Filter.antitone_seq_of_seq {α : Type u_1} (s : ℕ → Set α) : ∃ (t : ℕ → Set α), Antitone t ∧ ⨅ (i : ℕ), Filter.principal (s i) = ⨅ (i : ℕ), Filter.principal (t i)

theorem Filter.countable_biInf_eq_iInf_seq {α : Type u_1} {ι : Type u_4} [CompleteLattice α] {B : Set ι} (Bcbl : B.Countable) (Bne : B.Nonempty) (f : ι → α) : ∃ (x : ℕ → ι), ⨅ t ∈ B, f t = ⨅ (i : ℕ), f (x i)

theorem Filter.countable_biInf_eq_iInf_seq' {α : Type u_1} {ι : Type u_4} [CompleteLattice α] {B : Set ι} (Bcbl : B.Countable) (f : ι → α) {i₀ : ι} (h : f i₀ = ⊤) : ∃ (x : ℕ → ι), ⨅ t ∈ B, f t = ⨅ (i : ℕ), f (x i)

theorem Filter.countable_biInf_principal_eq_seq_iInf {α : Type u_1} {B : Set (Set α)} (Bcbl : B.Countable) : ∃ (x : ℕ → Set α), ⨅ t ∈ B, Filter.principal t = ⨅ (i : ℕ), Filter.principal (x i)

theorem Filter.HasAntitoneBasis.mem_iff {α : Type u_1} {ι : Type u_4} [Preorder ι] {l : Filter α} {s : ι → Set α} (hs : l.HasAntitoneBasis s) {t : Set α} : t ∈ l ↔ ∃ (i : ι), s i ⊆ t

theorem Filter.HasAntitoneBasis.mem {α : Type u_1} {ι : Type u_4} [Preorder ι] {l : Filter α} {s : ι → Set α} (hs : l.HasAntitoneBasis s) (i : ι) : s i ∈ l

theorem Filter.HasAntitoneBasis.hasBasis_ge {α : Type u_1} {ι : Type u_4} [Preorder ι] [IsDirected ι fun (x1 x2 : ι) => x1 ≤ x2] {l : Filter α} {s : ι → Set α} (hs : l.HasAntitoneBasis s) (i : ι) : l.HasBasis (fun (j : ι) => i ≤ j) s

theorem Filter.HasBasis.exists_antitone_subbasis {α : Type u_1} {ι' : Sort u_5} {f : Filter α} [h : f.IsCountablyGenerated] {p : ι' → Prop} {s : ι' → Set α} (hs : f.HasBasis p s) : ∃ (x : ℕ → ι'), (∀ (i : ℕ), p (x i)) ∧ f.HasAntitoneBasis fun (i : ℕ) => s (x i)

If f is countably generated and f.HasBasis p s, then f admits a decreasing basis enumerated by natural numbers such that all sets have the form s i. More precisely, there is a sequence i n such that p (i n) for all n and s (i n) is a decreasing sequence of sets which forms a basis of f

theorem Filter.exists_antitone_basis {α : Type u_1} (f : Filter α) [f.IsCountablyGenerated] : ∃ (x : ℕ → Set α), f.HasAntitoneBasis x

A countably generated filter admits a basis formed by an antitone sequence of sets.

theorem Filter.exists_antitone_seq {α : Type u_1} (f : Filter α) [f.IsCountablyGenerated] : ∃ (x : ℕ → Set α), Antitone x ∧ ∀ {s : Set α}, s ∈ f ↔ ∃ (i : ℕ), x i ⊆ s

instance Filter.Inf.isCountablyGenerated {α : Type u_1} (f : Filter α) (g : Filter α) [f.IsCountablyGenerated] [g.IsCountablyGenerated] : (f ⊓ g).IsCountablyGenerated

Equations

• ⋯ = ⋯

instance Filter.map.isCountablyGenerated {α : Type u_1} {β : Type u_2} (l : Filter α) [l.IsCountablyGenerated] (f : α → β) : (Filter.map f l).IsCountablyGenerated

Equations

• ⋯ = ⋯

instance Filter.comap.isCountablyGenerated {α : Type u_1} {β : Type u_2} (l : Filter β) [l.IsCountablyGenerated] (f : α → β) : (Filter.comap f l).IsCountablyGenerated

Equations

• ⋯ = ⋯

instance Filter.Sup.isCountablyGenerated {α : Type u_1} (f : Filter α) (g : Filter α) [f.IsCountablyGenerated] [g.IsCountablyGenerated] : (f ⊔ g).IsCountablyGenerated

Equations

• ⋯ = ⋯

instance Filter.prod.isCountablyGenerated {α : Type u_1} {β : Type u_2} (la : Filter α) (lb : Filter β) [la.IsCountablyGenerated] [lb.IsCountablyGenerated] : (la ×ˢ lb).IsCountablyGenerated

Equations

• ⋯ = ⋯

instance Filter.coprod.isCountablyGenerated {α : Type u_1} {β : Type u_2} (la : Filter α) (lb : Filter β) [la.IsCountablyGenerated] [lb.IsCountablyGenerated] : (la.coprod lb).IsCountablyGenerated

Equations

• ⋯ = ⋯

theorem Filter.isCountablyGenerated_seq {α : Type u_1} {ι' : Sort u_5} [Countable ι'] (x : ι' → Set α) : (⨅ (i : ι'), Filter.principal (x i)).IsCountablyGenerated

theorem Filter.isCountablyGenerated_of_seq {α : Type u_1} {f : Filter α} (h : ∃ (x : ℕ → Set α), f = ⨅ (i : ℕ), Filter.principal (x i)) : f.IsCountablyGenerated

theorem Filter.isCountablyGenerated_biInf_principal {α : Type u_1} {B : Set (Set α)} (h : B.Countable) : (⨅ s ∈ B, Filter.principal s).IsCountablyGenerated

theorem Filter.isCountablyGenerated_iff_exists_antitone_basis {α : Type u_1} {f : Filter α} : f.IsCountablyGenerated ↔ ∃ (x : ℕ → Set α), f.HasAntitoneBasis x

theorem Filter.isCountablyGenerated_principal {α : Type u_1} (s : Set α) : (Filter.principal s).IsCountablyGenerated

theorem Filter.isCountablyGenerated_pure {α : Type u_1} (a : α) : (pure a).IsCountablyGenerated

theorem Filter.isCountablyGenerated_bot {α : Type u_1} : ⊥.IsCountablyGenerated

theorem Filter.isCountablyGenerated_top {α : Type u_1} : ⊤.IsCountablyGenerated

instance Filter.iInf.isCountablyGenerated {ι : Sort u} {α : Type v} [Countable ι] (f : ι → Filter α) [∀ (i : ι), (f i).IsCountablyGenerated] : (⨅ (i : ι), f i).IsCountablyGenerated

Equations

• ⋯ = ⋯