# 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)