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 is said to be holomorphically separable, if whenever x ≠ y are two points in , there exists a holomorphic function , 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 are holomorphically separable, in particular, all domains in 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.