- Holomorphically separable
In
mathematics incomplex analysis , the concept of holomorphic separability is a measure forthe richness of the set ofholomorphic function s on acomplex manifold orcomplex space .Formal definition
A
complex manifold orcomplex space X is said to be holomorphically separable, if "x" ≠ "y" are two points in X, then there is aholomorphic function f in mathcal O(X), such that "f"("x") ≠ "f"("y").Often one says the
holomorphic function s "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 manifold s.
*A holomorphically complex manifold is not compact unless it is discrete and finite.
*The condition is part of the definition of aStein manifold .
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