# Irreducible analytic space

An 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. 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 $( X , {\mathcal O} )$ 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

 [1] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) [2] M. Hervé, "Several complex variables: local theory" , Oxford Univ. Press (1963)
How to Cite This Entry:
Irreducible analytic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irreducible_analytic_space&oldid=47432
This article was adapted from an original article by D.A. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article