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Holomorphic separability

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Source updated July 2, 2026

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.

01Formal 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.

02Usage 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.

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Sources and credits

This article is adapted from the Wikipedia article “Holomorphic separability”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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