# irreducible component

Let $G\subset {\u2102}^{N}$ be an open set.

###### Definition.

A locally analytic set (or an analytic variety) $V\subset G$ is said to be irreducible if whenever we have two locally analytic sets ${V}_{1}$ and ${V}_{2}$ such that $V={V}_{1}\cup {V}_{2}$, then either $V={V}_{1}$ or $V={V}_{2}$. Otherwise $V$ is
said to be . A maximal irreducible subvariety of $V$ is said to be an irreducible component^{} of $V$. Sometimes irreducible components are
called ircomps.

Note that if $V$ is an analytic variety in $G$, then a subvariety $W$ is an irreducible component of $V$ if and only if ${W}^{*}$ (the set of regular points^{} of $W$) is a connected complex analytic manifold. This means that the irreducible components of $V$ are the closures^{} of the connected components^{} of ${V}^{*}$.

## References

- 1 Hassler Whitney. . Addison-Wesley, Philippines, 1972.

Title | irreducible component |

Canonical name | IrreducibleComponent1 |

Date of creation | 2013-03-22 15:04:58 |

Last modified on | 2013-03-22 15:04:58 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 5 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32C25 |

Classification | msc 32A60 |

Synonym | ircomp |

Related topic | AnalyticSet |

Defines | irreducible analytic variety |

Defines | irreducible locally analytic set |

Defines | irreducible analytic variety |

Defines | reducible locally analytic set |

Defines | reducible analytic variety |