Holomorphic separability
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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.
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Formal definition
A complex manifold or complex space X {\displaystyle X} is said to be holomorphically separable, if whenever x ≠ y are two points in X {\displaystyle X} , there exists a holomorphic function f ∈ O ( X ) {\displaystyle f\in {\mathcal {O}}(X)} , such that f(x) ≠ f(y).cite-ref-1[1]
Often one says the holomorphic functions separate points.
Usage and examples
• All complex manifolds that can be mapped injectively into some C n {\displaystyle \mathbb {C} ^{n}} are holomorphically separable, in particular, all domains in C n {\displaystyle \mathbb {C} ^{n}} 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
• citerefkaupkaup2011Kaup, Ludger; Kaup, Burchard (9 May 2011). Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory. Walter de Gruyter. ISBN 9783110838350.
• citerefremmert1956Remmert, Reinhold (1956). "Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes". Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris (in French). 243: 118–121. Zbl 0070.30401.