Exhaustion of a domain
approximating sequence of domains
For a given domain in a topological space
, an exhaustion is a sequence of (in a certain sense regular) domains
such that
and
. For any domain
in a complex space
there exists an exhaustion by domains
that are, e.g., bounded by piecewise-smooth curves (in
) or by piecewise-smooth surfaces (in
,
). For any Riemann surface
there is a polyhedral exhaustion
, consisting of polyhedral domains
that are, each individually, connected unions of a finite number of triangles in a triangulation of
; moreover,
,
, and the boundary of each of the domains making up the open set
is, for sufficiently large
, just one of the boundary contours of
.
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 |
Exhaustion of a domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exhaustion_of_a_domain&oldid=17197