# irreducible component

Let $G\subset{\mathbb{C}}^{N}$ 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 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