# Exhaustion of a domain

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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