def IsIrreducibleComponent {X : Type u_1} [TopologicalSpace X] (s : Set X) : Prop

A subset s of a topological space is an irreducible component if it is closed, irreducible, and maximal among closed irreducible subsets containing it.

theorem isIrreducibleComponent_iff_mem {X : Type u_1} [TopologicalSpace X] (s : Set X) : IsIrreducibleComponent s ↔ s ∈ irreducibleComponents X

Our IsIrreducibleComponent predicate agrees with membership in Mathlib's irreducibleComponents X.

theorem IsIrreducibleComponent.isClosed {X : Type u_1} [TopologicalSpace X] {s : Set X} (h : IsIrreducibleComponent s) : IsClosed s

An irreducible component is closed.

theorem IsIrreducibleComponent.isIrreducible {X : Type u_1} [TopologicalSpace X] {s : Set X} (h : IsIrreducibleComponent s) : IsIrreducible s

An irreducible component is irreducible.

theorem IsIrreducibleComponent.eq_of_subset {X : Type u_1} [TopologicalSpace X] {s t : Set X} (h : IsIrreducibleComponent s) (htc : IsClosed t) (hti : IsIrreducible t) (hst : s ⊆ t) : t = s

Maximality of an irreducible component: any closed irreducible superset coincides with it.

theorem exists_isIrreducibleComponent_of_mem {X : Type u_1} [TopologicalSpace X] (x : X) : ∃ (s : Set X), IsIrreducibleComponent s ∧ x ∈ s

Every point of a topological space is contained in some irreducible component.