Holomorphically separable
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In mathematics in complex analysis, the concept of holomorphic separability is a measure for the richness of the set of holomorphic functions on a complex manifold or complex space.
[edit] 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 
, such that f(x) ≠ f(y).
Often one says the holomorphic functions separate points.
[edit] 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 complex manifold is not compact unless it is discrete and finite.
- The condition is part of the definition of a Stein manifold.
