Holomorphically separable

From Wikipedia, the free encyclopedia

In mathematics in complex analysis, the concept of holomorphic separability is a measure for the richness of the set of holomorphic functions on a complex manifold or complex space.

[edit] Formal definition

A complex manifold or complex space X is said to be holomorphically separable, if x ≠ y are two points in X, then there is a holomorphic function , such that f(x) ≠ f(y).

Often one says the holomorphic functions separate points.

[edit] Usage and examples

• All complex manifolds that can be mapped injectively into some are holomorphically separable, in particular, all domains in and all Stein manifolds.
• A holomorphically complex manifold is not compact unless it is discrete and finite.
• The condition is part of the definition of a Stein manifold.