Holomorphic separability

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

Formal definition

A complex manifold or complex space [math]\displaystyle{ X }[/math] is said to be holomorphically separable, if whenever x ≠ y are two points in [math]\displaystyle{ X }[/math], there exists a holomorphic function [math]\displaystyle{ f \in \mathcal O(X) }[/math], such that f(x) ≠ f(y).

Often one says the holomorphic functions separate points.

Usage and examples

• All complex manifolds that can be mapped injectively into some [math]\displaystyle{ \mathbb{C}^n }[/math] are holomorphically separable, in particular, all domains in [math]\displaystyle{ \mathbb{C}^n }[/math] and all Stein manifolds.
• A holomorphically separable complex manifold is not compact unless it is discrete and finite.
• The condition is part of the definition of a Stein manifold.