Definition:Irreducible Component

Definition
Let $T = \struct {S, \tau}$ be a topological space.

A subset $Y \subseteq S$ is an irreducible component of $T$ :
 * $Y$ is irreducible
 * $Y$ is not a proper subset of an irreducible subset of $S$.

That is, :
 * $Y$ is maximal in the ordered set of irreducible subsets of $S$, ordered by the subset relation.

Also known as
Equivalently, we also say:
 * $Y$ is an irreducible component of $S$

or:
 * the subspace $\struct {Y, \tau_Y}$ is an irreducible component of $T$.

Also see

 * Point is Contained in Irreducible Component
 * Irreducible Component is Closed