---
tag: SY2MWC434Z
kind: concept
name: Irreducible component
msc: [14A10, 14A15]
sources:
  - url: https://en.wikipedia.org/wiki/Irreducible_component
same_as:
  wikidata: Q6073574
  wikipedia: Irreducible component
---

## Summary

A topological space is irreducible when it is nonempty and is not the union of two proper closed subsets.
An irreducible component of a space is a maximal irreducible subset.
In a Noetherian space, such as an algebraic set with its Zariski topology, there are only finitely many irreducible components.
The space is their union.
This splits every algebraic set into irreducible pieces, the varieties.

## Explanations

### L1 human

An algebraic set can be made of several pieces that cannot be split further.
The set of solutions of $xy = 0$ in the plane is the union of the two coordinate axes.
Each axis is a single piece that is not a union of two smaller algebraic sets, and the two axes are the irreducible components.
A finite set of two or more points is not irreducible: each point is a piece of its own.