Documentation

Mathlib.Order.Filter.Ker

Kernel of a filter #

In this file we define the kernel Filter.ker f of a filter f to be the intersection of all its sets.

We also prove that Filter.principal and Filter.ker form a Galois coinsertion and prove other basic theorems about Filter.ker.

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theorem Filter.ker_def {α : Type u_2} (f : Filter α) :
f.ker = ⋂ s ∈ f, s
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@[simp]
theorem Filter.mem_ker {α : Type u_2} {f : Filter α} {a : α} :
a ∈ f.ker ↔ ∀ s ∈ f, a ∈ s
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@[simp]
theorem Filter.subset_ker {α : Type u_2} {f : Filter α} {s : Set α} :
s ⊆ f.ker ↔ ∀ t ∈ f, s ⊆ t
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def Filter.gi_principal_ker {α : Type u_2} :
GaloisCoinsertion principal ker

Filter.principal forms a Galois coinsertion with Filter.ker.

Equations
Instances For
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    theorem Filter.ker_mono {α : Type u_2} :
    Monotone ker
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    theorem Filter.ker_surjective {α : Type u_2} :
    Function.Surjective ker
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    @[simp]
    theorem Filter.ker_bot {α : Type u_2} :
    ⊥.ker = ∅
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    @[simp]
    theorem Filter.ker_top {α : Type u_2} :
    ⊤.ker = Set.univ
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    @[simp]
    theorem Filter.ker_eq_univ {α : Type u_2} {f : Filter α} :
    f.ker = Set.univ ↔ f = ⊤
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    @[simp]
    theorem Filter.ker_inf {α : Type u_2} (f g : Filter α) :
    (f ⊓ g).ker = f.ker ∩ g.ker
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    @[simp]
    theorem Filter.ker_iInf {ι : Sort u_1} {α : Type u_2} (f : ι → Filter α) :
    (⨅ (i : ι), f i).ker = ⋂ (i : ι), (f i).ker
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    @[simp]
    theorem Filter.ker_sInf {α : Type u_2} (S : Set (Filter α)) :
    (sInf S).ker = ⋂ f ∈ S, f.ker
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    @[simp]
    theorem Filter.ker_principal {α : Type u_2} (s : Set α) :
    (principal s).ker = s
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    @[simp]
    theorem Filter.ker_pure {α : Type u_2} (a : α) :
    (pure a).ker = {a}
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    @[simp]
    theorem Filter.ker_comap {α : Type u_2} {β : Type u_3} (m : α → β) (f : Filter β) :
    (comap m f).ker = m ⁻¹' f.ker
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    @[simp]
    theorem Filter.ker_iSup {ι : Sort u_1} {α : Type u_2} (f : ι → Filter α) :
    (⨆ (i : ι), f i).ker = ⋃ (i : ι), (f i).ker
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    @[simp]
    theorem Filter.ker_sSup {α : Type u_2} (S : Set (Filter α)) :
    (sSup S).ker = ⋃ f ∈ S, f.ker
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    @[simp]
    theorem Filter.ker_sup {α : Type u_2} (f g : Filter α) :
    (f ⊔ g).ker = f.ker ∪ g.ker
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    @[simp]
    theorem Filter.ker_prod {α : Type u_2} {β : Type u_3} (f : Filter α) (g : Filter β) :
    (f ×ˢ g).ker = f.ker ×ˢ g.ker
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    @[simp]
    theorem Filter.ker_pi {ι : Type u_4} {α : ι → Type u_5} (f : (i : ι) → Filter (α i)) :
    (pi f).ker = Set.univ.pi fun (i : ι) => (f i).ker