Namespaces
Variants
Actions

Difference between revisions of "Exhaustion of a domain"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
 +
<!--
 +
e0368501.png
 +
$#A+1 = 20 n = 0
 +
$#C+1 = 20 : ~/encyclopedia/old_files/data/E036/E.0306850 Exhaustion of a domain,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''approximating sequence of domains''
 
''approximating sequence of domains''
  
For a given domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e0368501.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e0368502.png" />, an exhaustion is a sequence of (in a certain sense regular) domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e0368503.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e0368504.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e0368505.png" />. For any domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e0368506.png" /> in a complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e0368507.png" /> there exists an exhaustion by domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e0368508.png" /> that are, e.g., bounded by piecewise-smooth curves (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e0368509.png" />) or by piecewise-smooth surfaces (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e03685010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e03685011.png" />). For any Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e03685012.png" /> there is a polyhedral exhaustion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e03685013.png" />, consisting of polyhedral domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e03685014.png" /> that are, each individually, connected unions of a finite number of triangles in a [[Triangulation|triangulation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e03685015.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e03685016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e03685017.png" />, and the boundary of each of the domains making up the open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e03685018.png" /> is, for sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e03685019.png" />, just one of the boundary contours of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036850/e03685020.png" />.
+
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|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Stoilov,  "The theory of functions of a complex variable" , '''2''' , Moscow  (1962)  pp. Chapt. 5  (In Russian; translated from Rumanian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Stoilov,  "The theory of functions of a complex variable" , '''2''' , Moscow  (1962)  pp. Chapt. 5  (In Russian; translated from Rumanian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 19:38, 5 June 2020


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=46873
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article