Definition:Irreducible Component

Definition

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

A subset $Y \subseteq S$ is an irreducible component of $T$ if and only if:

$Y$ is irreducible

$Y$ is not a proper subset of an irreducible subset of $S$.

That is, if and only if:

$Y$ 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