# Covering domain

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domain over $\mathbf{C}^n$

A pair $(X,\pi)$, where $X$ is an arcwise-connected Hausdorff space and $\pi$ is a local homeomorphism, called a projection. Covering domains are encountered in the analytic continuation of holomorphic functions. For every analytic (possibly multivalent) function $f$ in a domain $D \subset \mathbf{C}^n$ there is a corresponding covering domain $\tilde D$ with a projection $\pi : \tilde D \rightarrow D$, just as for every analytic function of one complex variable there is a corresponding Riemann surface; the function $f$ is single-valued on $\tilde D$. Covering domains are also called Riemann domains.

#### References

 [1] B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)

A covering domain is sometimes called a manifold spread over $\mathbf{C}^n$. See also Domain of holomorphy; Riemannian domain; Holomorphic envelope.