Exhaustion of a domain

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