# Definition:Irreducible Component

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## 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 $T$**.