irreducible component

Let $G\subset {ℂ}^N$ 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 $V$ issaid to be . A maximal irreducible subvariety of $V$ is said to be an irreducible component of $V$. Sometimes irreducible components arecalled 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 $V^{*}$.