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A property of holomorphic functions on domains in the complex plane: the set of values of every non-constant holomorphic function on a domain  $  D \subset  \mathbf C $
is also a domain, that is, an open connected set. The basic property here is that of openness of the image, which follows from Rouché's theorem or from the principle of the argument (cf. [[Rouché theorem|Rouché theorem]]; [[Argument, principle of the|Argument, principle of the]]). The principle of preservation of domain can be considered as a generalization of the [[Maximum-modulus principle|maximum-modulus principle]] for holomorphic functions.

The principle of preservation of domain holds for holomorphic functions on an arbitrary complex manifold: the set of values of any non-constant holomorphic function on a connected [[Complex manifold|complex manifold]]  $  X $
is a domain in the complex plane. It also holds for holomorphic mappings from complex manifolds into Riemann surfaces. However, holomorphic mappings  $  f :  X \rightarrow Y $
into complex manifolds  $  Y $
of dimension exceeding 1 need not be open, in general: If  $  f $
is non-constant, but has rank everywhere less than  $   \mathop{\rm dim}  Y $, 
say, then the image  $  f ( X) \subset  Y $
has no interior points, in general. Openness can also be violated in the case when  $   \mathop{\rm rank}  f <  \mathop{\rm dim}  Y $
on sets of small dimension. For example, under the mapping

$$ 
( z _ {1} , z _ {2} )  \rightarrow  ( z _ {1} , z _ {1} z _ {2} )
$$

from  $  \mathbf C  ^ {2} $
into itself, the image is the non-open set  $  \mathbf C  ^ {2} \setminus  \{ w _ {1} = 0;   w _ {2} \neq 0 \} $. 
The principle of preservation of domain holds for holomorphic mappings if the condition that  $  f $
be non-constant is replaced by stronger requirements, one of the simplest being the zero-dimensionality of the set of points for which  $   \mathop{\rm rank}  f <  \mathop{\rm dim}  Y $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,   "Einführung in die Funktionentheorie" , '''1–3''' , Teubner  (1965–1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.C. Gunning,   H. Rossi,   "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR></table>

====Comments====
In the older German literature the concept occurs as the  "GebietstreueGebietstreue"  of a holomorphic mapping. In English, one usually speaks of the open mapping principle, cf. [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.T. Whyburn,   "Topological analysis" , Princeton Univ. Press  (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.M. Chirka,   "Complex analytic sets" , Kluwer  (1989)  (Translated from Russian)</TD></TR></table>