Holomorphic separability

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 xy are two points in , 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 into some are holomorphically separable, in particular, all domains in 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.
gollark: Ale: because that's what my `python-obfuscate` script uses.P.S. curse slowmode.
gollark: No, I mean I doubt they support the slight insanity that snippet uses.
gollark: ... probably? But I doubt they support this.
gollark: (That's Python, though)
gollark: ```pythonimport zlib,base64,marshal;exec(marshal.loads(zlib.decompress(base64.b85decode("c$`aSKmtra>;S~YJU}9qA%&rtk&z*VF_=M<sfsHkH95mMzo<aL)K8P~7HdIKW?sokh9X9wQZVt0!=|_dD41?{i^C>2KczG$)ea>78Dug5Xm=Gu"))))```
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