Theorems · Theorem · general topology

TopologicalSpace.NoetherianSpace.exists_finite_set_isClosed_irreducible

∀ {α : Type u_1} [inst : TopologicalSpace α] [TopologicalSpace.NoetherianSpace α] {s : Set α},
  IsClosed s → ∃ S, S.Finite ∧ (∀ t ∈ S, IsClosed t) ∧ (∀ t ∈ S, IsIrreducible t) ∧ s = ⋃₀ S

In a Noetherian space, every closed set is a finite union of irreducible closed sets.

Defined in

Mathlib.Topology.NoetherianSpace

Cited by

1 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace.NoetherianSpace