Holomorphically separable

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

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 $f\; in\; mathcal\; O(X)$, 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 complex manifold is not compact unless it is discrete and finite.
*The condition is part of the definition of a Stein manifold.