- Source: 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
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).
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
Kata Kunci Pencarian:
- Holomorphic separability
- Holomorphic function
- Function of several complex variables
- Complex geometry
- Stein manifold
- Hilbert space
- Equicontinuity
- Dirac delta function
- Oswald Veblen Prize in Geometry
- Glossary of number theory
