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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.
==Formal definition== A complex manifold or complex space <math>X</math> is said to be holomorphically separable, if whenever ''x'' ≠ ''y'' are two points in <math>X</math>, there exists a holomorphic function <math>f \in \mathcal O(X)</math>, such that ''f''(''x'') ≠ ''f''(''y'').<ref>{{Cite book |last=Grauert |first=Hans |author-link=Hans Grauert |title=Theory of Stein Spaces |last2=Remmert |first2=Reinhold |author-link2=Reinhold Remmert |publisher=Springer-Verlag |year=2004 |isbn=3-540-00373-8 |edition=Reprint of the 1979 |publication-date=2004 |pages=117 |translator-last=Huckleberry |translator-first=Alan}}</ref>
Often one says the holomorphic functions ''separate points''.
==Usage and examples== *All complex manifolds that can be mapped injectively into some <math>\mathbb{C}^n</math> are holomorphically separable, in particular, all domains in <math>\mathbb{C}^n</math> 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 == {{refbegin}} *{{cite book |isbn=9783110838350|title=Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory|last1=Kaup|first1=Ludger|last2=Kaup|first2=Burchard|date=9 May 2011|publisher=Walter de Gruyter |url={{Google books|4YVXCgewhTIC|Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory|plainurl=yes}}}} *{{cite journal|jstor=2034564 |doi=10.1090/S0002-9939-1960-0170034-8|title=Holomorphic mappings of complex spaces|year=1960|last1=Narasimhan|first1=Raghavan|journal=Proceedings of the American Mathematical Society|volume=11|issue=5|pages=800–804|doi-access=free}} *{{cite journal |arxiv=1108.2078|last1=Noguchi|first1=Junjiro|title=Another Direct Proof of Oka's Theorem (Oka IX)|year=2011|mr=3086750|journal=J. Math. Sci. Univ. Tokyo|url=https://www.ms.u-tokyo.ac.jp/journal/pdf/jms190407.pdf|volume=19|issue=4}} *{{cite journal |title =Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes |journal= Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris| pages=118–121| last1=Remmert|first1=Reinhold |author-link=Reinhold Remmert|year=1956|volume=243|url=https://gallica.bnf.fr/ark:/12148/bpt6k3195v/f118.image.r|language=fr|zbl=0070.30401}} {{refend}}
{{DEFAULTSORT:Holomorphically Separable}} Category:Complex analysis Category:Several complex variables