# Covering domain

*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) |

#### Comments

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

#### References

[a1] | R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) pp. Chapt. 1, Section G |

[a2] | H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1979) (Translated from German) |

**How to Cite This Entry:**

Covering domain.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Covering_domain&oldid=49715