File: Mathlib/Topology/NoetherianSpace.lean

full_name: TopologicalSpace.NoetherianSpace.wellFounded_opens
code: /-- Type class for noetherian spaces. It is defined to be spaces whose open sets satisfies ACC. -/
@[mk_iff]
class NoetherianSpace : Prop where
  wellFounded_opens : WellFounded ((· > ·) : Opens α → Opens α → Prop)
goal: WellFounded fun x x_1 => x > x_1

full_name: TopologicalSpace.noetherianSpace_iff
code: /-- Type class for noetherian spaces. It is defined to be spaces whose open sets satisfies ACC. -/
@[mk_iff]
class NoetherianSpace : Prop where
  wellFounded_opens : WellFounded ((· > ·) : Opens α → Opens α → Prop)
goal: TopologicalSpace.NoetherianSpace α ↔ WellFounded fun x x_1 => x > x_1

full_name: TopologicalSpace.NoetherianSpace
code: /-- Type class for noetherian spaces. It is defined to be spaces whose open sets satisfies ACC. -/
@[mk_iff]
class NoetherianSpace : Prop where
  wellFounded_opens : WellFounded ((· > ·) : Opens α → Opens α → Prop)
goal: Prop

full_name: TopologicalSpace.noetherianSpace_iff_opens
code: theorem noetherianSpace_iff_opens : NoetherianSpace α ↔ ∀ s : Opens α, IsCompact (s : Set α) 
goal: TopologicalSpace.NoetherianSpace α ↔ ∀ (s : TopologicalSpace.Opens α), IsCompact ↑s

full_name: TopologicalSpace.NoetherianSpace.compactSpace
code: instance (priority := 100) NoetherianSpace.compactSpace [h : NoetherianSpace α] : CompactSpace α 
goal: CompactSpace α

full_name: TopologicalSpace.NoetherianSpace.isCompact
code: /-- In a Noetherian space, all sets are compact. -/
protected theorem NoetherianSpace.isCompact [NoetherianSpace α] (s : Set α) : IsCompact s 
goal: IsCompact s

full_name: Inducing.noetherianSpace
code: protected theorem _root_.Inducing.noetherianSpace [NoetherianSpace α] {i : β → α}
    (hi : Inducing i) : NoetherianSpace β 
goal: TopologicalSpace.NoetherianSpace β

full_name: TopologicalSpace.NoetherianSpace.set
code: /-- [Stacks: Lemma 0052 (1)](https://stacks.math.columbia.edu/tag/0052)-/
instance NoetherianSpace.set [NoetherianSpace α] (s : Set α) : NoetherianSpace s 
goal: TopologicalSpace.NoetherianSpace ↑s

full_name: TopologicalSpace.noetherianSpace_TFAE
code: theorem noetherianSpace_TFAE :
    TFAE [NoetherianSpace α,
      WellFounded fun s t : Closeds α => s < t,
      ∀ s : Set α, IsCompact s,
      ∀ s : Opens α, IsCompact (s : Set α)] 
goal: [TopologicalSpace.NoetherianSpace α, WellFounded fun s t => s < t, ∀ (s : Set α), IsCompact s,
    ∀ (s : TopologicalSpace.Opens α), IsCompact ↑s].TFAE

full_name: TopologicalSpace.noetherianSpace_iff_isCompact
code: theorem noetherianSpace_iff_isCompact : NoetherianSpace α ↔ ∀ s : Set α, IsCompact s 
goal: TopologicalSpace.NoetherianSpace α ↔ ∀ (s : Set α), IsCompact s

full_name: TopologicalSpace.NoetherianSpace.wellFounded_closeds
code: theorem NoetherianSpace.wellFounded_closeds [NoetherianSpace α] :
    WellFounded fun s t : Closeds α => s < t 
goal: WellFounded fun s t => s < t

full_name: TopologicalSpace.instNoetherianSpaceCofiniteTopology
code: instance {α} : NoetherianSpace (CofiniteTopology α) 
goal: TopologicalSpace.NoetherianSpace (CofiniteTopology α)

full_name: TopologicalSpace.noetherianSpace_of_surjective
code: theorem noetherianSpace_of_surjective [NoetherianSpace α] (f : α → β) (hf : Continuous f)
    (hf' : Function.Surjective f) : NoetherianSpace β 
goal: TopologicalSpace.NoetherianSpace β

full_name: TopologicalSpace.noetherianSpace_iff_of_homeomorph
code: theorem noetherianSpace_iff_of_homeomorph (f : α ≃ₜ β) : NoetherianSpace α ↔ NoetherianSpace β 
goal: TopologicalSpace.NoetherianSpace α ↔ TopologicalSpace.NoetherianSpace β

full_name: TopologicalSpace.NoetherianSpace.range
code: theorem NoetherianSpace.range [NoetherianSpace α] (f : α → β) (hf : Continuous f) :
    NoetherianSpace (Set.range f) 
goal: TopologicalSpace.NoetherianSpace ↑(Set.range f)

full_name: TopologicalSpace.noetherianSpace_set_iff
code: theorem noetherianSpace_set_iff (s : Set α) :
    NoetherianSpace s ↔ ∀ t, t ⊆ s → IsCompact t 
goal: TopologicalSpace.NoetherianSpace ↑s ↔ ∀ t ⊆ s, IsCompact t

full_name: TopologicalSpace.noetherian_univ_iff
code: @[simp]
theorem noetherian_univ_iff : NoetherianSpace (Set.univ : Set α) ↔ NoetherianSpace α 
goal: TopologicalSpace.NoetherianSpace ↑Set.univ ↔ TopologicalSpace.NoetherianSpace α

full_name: TopologicalSpace.NoetherianSpace.iUnion
code: theorem NoetherianSpace.iUnion {ι : Type*} (f : ι → Set α) [Finite ι]
    [hf : ∀ i, NoetherianSpace (f i)] : NoetherianSpace (⋃ i, f i) 
goal: TopologicalSpace.NoetherianSpace ↑(⋃ i, f i)

full_name: TopologicalSpace.NoetherianSpace.discrete
code: theorem NoetherianSpace.discrete [NoetherianSpace α] [T2Space α] : DiscreteTopology α 
goal: DiscreteTopology α

full_name: TopologicalSpace.NoetherianSpace.finite
code: /-- Spaces that are both Noetherian and Hausdorff are finite. -/
theorem NoetherianSpace.finite [NoetherianSpace α] [T2Space α] : Finite α 
goal: Finite α

full_name: TopologicalSpace.Finite.to_noetherianSpace
code: instance (priority := 100) Finite.to_noetherianSpace [Finite α] : NoetherianSpace α 
goal: TopologicalSpace.NoetherianSpace α

full_name: TopologicalSpace.NoetherianSpace.exists_finite_set_closeds_irreducible
code: /-- In a Noetherian space, every closed set is a finite union of irreducible closed sets. -/
theorem NoetherianSpace.exists_finite_set_closeds_irreducible [NoetherianSpace α] (s : Closeds α) :
    ∃ S : Set (Closeds α), S.Finite ∧ (∀ t ∈ S, IsIrreducible (t : Set α)) ∧ s = sSup S 
goal: ∃ S, S.Finite ∧ (∀ t ∈ S, IsIrreducible ↑t) ∧ s = sSup S

full_name: TopologicalSpace.NoetherianSpace.exists_finite_set_isClosed_irreducible
code: /-- In a Noetherian space, every closed set is a finite union of irreducible closed sets. -/
theorem NoetherianSpace.exists_finite_set_isClosed_irreducible [NoetherianSpace α]
    {s : Set α} (hs : IsClosed s) : ∃ S : Set (Set α), S.Finite ∧
      (∀ t ∈ S, IsClosed t) ∧ (∀ t ∈ S, IsIrreducible t) ∧ s = ⋃₀ S 
goal: ∃ S, S.Finite ∧ (∀ t ∈ S, IsClosed t) ∧ (∀ t ∈ S, IsIrreducible t) ∧ s = ⋃₀ S

full_name: TopologicalSpace.NoetherianSpace.exists_finset_irreducible
code: /-- In a Noetherian space, every closed set is a finite union of irreducible closed sets. -/
theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Closeds α) :
    ∃ S : Finset (Closeds α), (∀ k : S, IsIrreducible (k : Set α)) ∧ s = S.sup id 
goal: ∃ S, (∀ (k : { x // x ∈ S }), IsIrreducible ↑↑k) ∧ s = S.sup id

full_name: TopologicalSpace.NoetherianSpace.finite_irreducibleComponents
code: /-- [Stacks: Lemma 0052 (2)](https://stacks.math.columbia.edu/tag/0052) -/
theorem NoetherianSpace.finite_irreducibleComponents [NoetherianSpace α] :
    (irreducibleComponents α).Finite 
goal: (irreducibleComponents α).Finite

full_name: TopologicalSpace.NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent
code: /-- [Stacks: Lemma 0052 (3)](https://stacks.math.columbia.edu/tag/0052) -/
theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [NoetherianSpace α]
    (Z : Set α) (H : Z ∈ irreducibleComponents α) :
    ∃ o : Set α, IsOpen o ∧ o ≠ ∅ ∧ o ≤ Z 
goal: ∃ o, IsOpen o ∧ o ≠ ∅ ∧ o ≤ Z