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 $f\in {𝒪}(X)$, 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.