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, 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 are holomorphically separable, in particular, all domains 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.

References

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