theorem noetherian_finite_union_irreducible_components (X : Type u_1) [TopologicalSpace X] [TopologicalSpace.NoetherianSpace X] : (irreducibleComponents X).Finite ∧ ⋃₀ irreducibleComponents X = Set.univ

Lecture 4, Proposition 5: a Noetherian topological space has finitely many irreducible components, whose union is the whole space.

theorem noetherian_finite_union_closed_irreducible (X : Type u_1) [TopologicalSpace X] [TopologicalSpace.NoetherianSpace X] : ∃ (S : Set (Set X)), S.Finite ∧ (∀ t ∈ S, IsClosed t) ∧ (∀ t ∈ S, IsIrreducible t) ∧ ⋃₀ S = Set.univ

Existential form of Proposition 5: a Noetherian topological space admits some finite covering by closed irreducible subsets.

theorem vanishingIdeal_sUnion_finset {R : Type u_1} [CommRing R] (T : Finset (Set (PrimeSpectrum R))) : PrimeSpectrum.vanishingIdeal (⋃₀ ↑T) = T.inf fun (x : Set (PrimeSpectrum R)) => PrimeSpectrum.vanishingIdeal x

The vanishing ideal of a finite union of subsets of Spec R is the infimum of the individual vanishing ideals.

theorem radical_ideal_eq_finite_iInf_primes {R : Type u_1} [CommRing R] [IsNoetherianRing R] (I : Ideal R) (hI : I.IsRadical) : ∃ (S : Finset (Ideal R)), (∀ P ∈ S, P.IsPrime) ∧ I = S.inf id

Ring-theoretic consequence of Proposition 5: in a Noetherian ring, every radical ideal is a finite intersection of primes.

theorem closed_irreducible_iff_vanishingIdeal_prime {R : Type u_1} [CommRing R] {Z : Set (PrimeSpectrum R)} (_hZ : IsClosed Z) : IsIrreducible Z ↔ (PrimeSpectrum.vanishingIdeal Z).IsPrime

A closed subset of Spec R is irreducible iff its vanishing ideal is prime.