View source for 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====
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<TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Shabat,   "Introduction of complex analysis" , '''2''' , Moscow  (1976)  (In Russian)</TD></TR>
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====Comments====
A covering domain is sometimes called a manifold spread over $\mathbf{C}^n$. See also [[Domain of holomorphy]]; [[Riemannian domain]]; [[Holomorphic envelope]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Gunning,   H. Rossi,   "Analytic functions of several complex variables" , Prentice-Hall  (1965)  pp. Chapt. 1, Section G</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Grauert,   R. Remmert,   "Theory of Stein spaces" , Springer  (1979)  (Translated from German)</TD></TR>
</table>

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