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Difference between revisions of "Exhaustion of a domain"

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For a given domain  $  D $
 
For a given domain  $  D $
 
in a topological space  $  X $,  
 
in a topological space  $  X $,  
an exhaustion is a sequence of (in a certain sense regular) domains  $  \{ D _ {k} \} _ {k=} ^  \infty  \subset  D $
+
an exhaustion is a sequence of (in a certain sense regular) domains  $  \{ D _ {k} \} _ {k=1}^  \infty  \subset  D $
such that  $  \overline{D}\; _ {k} \subset  D _ {k+} 1 $
+
such that  $  \overline{D}\; _ {k} \subset  D _ {k+1} $
and  $  \cup _ {k=} ^  \infty  D _ {k} = D $.  
+
and  $  \cup _ {k=1}^  \infty  D _ {k} = D $.  
 
For any domain  $  D $
 
For any domain  $  D $
 
in a complex space  $  \mathbf C  ^ {n} $
 
in a complex space  $  \mathbf C  ^ {n} $
Line 25: Line 25:
 
$  n > 1 $).  
 
$  n > 1 $).  
 
For any Riemann surface  $  S $
 
For any Riemann surface  $  S $
there is a polyhedral exhaustion  $  \{ \Pi _ {k} \} _ {k=} ^  \infty  $,  
+
there is a polyhedral exhaustion  $  \{ \Pi _ {k} \} _{k=1}^  \infty  $,  
 
consisting of polyhedral domains  $  \Pi _ {k} $
 
consisting of polyhedral domains  $  \Pi _ {k} $
 
that are, each individually, connected unions of a finite number of triangles in a [[Triangulation|triangulation]] of  $  S $;  
 
that are, each individually, connected unions of a finite number of triangles in a [[Triangulation|triangulation]] of  $  S $;  
moreover,  $  \overline \Pi \; _ {k} \subset  \Pi _ {k+} 1 $,  
+
moreover,  $  \overline \Pi \; _ {k} \subset  \Pi _ {k+1} $,  
$  \cup _ {k=} ^  \infty  \Pi _ {k} = S $,  
+
$  \cup _{k=1}^  \infty  \Pi _ {k} = S $,  
 
and the boundary of each of the domains making up the open set  $  S \setminus  \overline \Pi \; _ {k} $
 
and the boundary of each of the domains making up the open set  $  S \setminus  \overline \Pi \; _ {k} $
 
is, for sufficiently large  $  k $,  
 
is, for sufficiently large  $  k $,  

Latest revision as of 19:20, 17 January 2024


approximating sequence of domains

For a given domain $ D $ in a topological space $ X $, an exhaustion is a sequence of (in a certain sense regular) domains $ \{ D _ {k} \} _ {k=1}^ \infty \subset D $ such that $ \overline{D}\; _ {k} \subset D _ {k+1} $ and $ \cup _ {k=1}^ \infty D _ {k} = D $. For any domain $ D $ in a complex space $ \mathbf C ^ {n} $ there exists an exhaustion by domains $ D _ {k} $ that are, e.g., bounded by piecewise-smooth curves (in $ \mathbf C ^ {1} $) or by piecewise-smooth surfaces (in $ \mathbf C ^ {n} $, $ n > 1 $). For any Riemann surface $ S $ there is a polyhedral exhaustion $ \{ \Pi _ {k} \} _{k=1}^ \infty $, consisting of polyhedral domains $ \Pi _ {k} $ that are, each individually, connected unions of a finite number of triangles in a triangulation of $ S $; moreover, $ \overline \Pi \; _ {k} \subset \Pi _ {k+1} $, $ \cup _{k=1}^ \infty \Pi _ {k} = S $, and the boundary of each of the domains making up the open set $ S \setminus \overline \Pi \; _ {k} $ is, for sufficiently large $ k $, just one of the boundary contours of $ \Pi _ {k} $.

References

[1] S. Stoilov, "The theory of functions of a complex variable" , 2 , Moscow (1962) pp. Chapt. 5 (In Russian; translated from Rumanian)

Comments

The fact that any pseudo-convex domain (cf. Pseudo-convex and pseudo-concave) can be exhausted by smooth, strictly pseudo-convex domains is of fundamental importance in higher-dimensional complex analysis, cf. [a2].

References

[a1] G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10
[a2] L.V. Ahlfors, L. Sario, "Riemann surfaces" , Princeton Univ. Press (1960) pp. Chapt. 1
[a3] S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) pp. Sect. 7.1
How to Cite This Entry:
Exhaustion of a domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exhaustion_of_a_domain&oldid=55173
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article