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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def Filter.ker {α : Type u_2} (f : Filter α) :
Set α

The kernel of a filter is the intersection of all its sets.

Equations
Instances For
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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 Filter.principal Filter.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 Filter.ker
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      theorem Filter.ker_surjective {α : Type u_2} :
      Function.Surjective Filter.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 : Filter α) (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) :
      (Filter.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) :
      (Filter.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 : Filter α) (g : Filter α) :
      (f ⊔ g).ker = f.ker ∪ g.ker