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 $$X$$ is said to be holomorphically separable, if whenever x ≠ y are two points in $$X$$, there exists a holomorphic function $$f \in \mathcal O(X)$$, 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 $$\mathbb{C}^n$$ are holomorphically separable, in particular, all domains in $$\mathbb{C}^n$$ 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.