View source for Irreducible analytic space

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An [[Analytic space|analytic space]] that cannot be represented as the union of a locally finite family of analytic subspaces. An irreducible analytic space is a generalization of the concept of an irreducible [[Analytic set|analytic set]]. Every analytic space can be represented uniquely as an irreducible union of a locally finite family of irreducible analytic subspaces, its so-called irreducible components (the stratification of a space into irreducible components). A complex-analytic manifold is irreducible if and only if it is connected; the irreducible components of a manifold are its connected components. The germ of an analytic space at a given point of it is called irreducible if it cannot be represented as a union of finitely many germs of analytic subspaces at the same point. Every germ of an analytic space at a point can be represented uniquely as a union of finitely many irreducible subgerms of it. The germ of a reduced [[Complex space|complex space]]  $  ( X , {\mathcal O} ) $
at a point  $  x \in X $
is irreducible if and only if the local ring  $  {\mathcal O} _ {x} $
has no divisors of zero. A complex space whose germs at all its points are irreducible is itself irreducible if and only if it is connected; the irreducible components of a complex space are its connected components.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.C. Gunning,   H. Rossi,   "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Hervé,   "Several complex variables: local theory" , Oxford Univ. Press  (1963)</TD></TR></table>