Holomorphic separability
title: "Holomorphic separability" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["complex-analysis", "several-complex-variables"] topic_path: "general/complex-analysis" source: "https://en.wikipedia.org/wiki/Holomorphic_separability" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
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 is said to be holomorphically separable, if whenever x ≠ y are two points in X, there exists a holomorphic function 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 \mathbb{C}^n are holomorphically separable, in particular, all domains in \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
References
- Grauert, Hans. (2004). "Theory of Stein Spaces". Springer-Verlag.
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