irreducible componentIrreducibleComponent22005-02-22 20:10:562005-02-23 00:13:55Definitionirreducible analytic varietyirreducible locally analytic setirreducible analytic varietyreducible locally analytic setreducible analytic variety% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{defn}{Definition}
\theoremstyle{theorem}
\newtheorem*{thm}{Theorem}Let $G \subset {\mathbb{C}}^N$ be an open set.
\begin{defn}
A locally analytic set (or an analytic variety) $V \subset G$ is said to be {\em 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 {\em \PMlinkescapetext{reducible}}. A maximal irreducible subvariety of $V$ is said to be an {\em irreducible component} of $V$. Sometimes irreducible components are
called {\em ircomps}.
\end{defn}
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 $V^*$.
\begin{thebibliography}{9}
\bibitem{Whitney:varieties}
Hassler Whitney.
{\em \PMlinkescapetext{Complex Analytic Varieties}}.
Addison-Wesley, Philippines, 1972.
\end{thebibliography}