# 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) |

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Irreducible analytic space.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Irreducible_analytic_space&oldid=47432