# Domain

A non-empty connected open set in a topological space $X$. The closure $\overline{D}\;$ of a domain $D$ is called a closed domain; the closed set $\textrm{ Fr } D = \overline{D}\; \setminus D$ is called the boundary of $D$. The points $x \in D$ are also called the interior points of $D$; the points $x \in \textrm{ Fr } D$ are called the boundary points of $D$; the points of the complement $C \overline{D}\; = X \setminus \overline{D}\;$ are called the exterior points of $D$.

Any two points of a domain $D$ in the real Euclidean space $\mathbf R ^ {n}$, $n \geq 1$( or in the complex space $\mathbf C ^ {m}$, $m \geq 1$, or on a Riemann surface or in a Riemannian domain), can be joined by a path (or arc) lying completely in $D$; if $D \subset \mathbf R ^ {n}$ or $D \subset \mathbf C ^ {m}$, they can even be joined by a polygonal path with a finite number of edges. Finite and infinite open intervals are the only domains in the real line $\mathbf R = \mathbf R ^ {1}$; their boundaries consist of at most two points. A domain $D$ in the plane is called simply connected if any closed path in $D$ can be continuously deformed to a point, remaining throughout in $D$. In general, the boundary of a simply-connected domain in the (open) plane $\mathbf R ^ {2}$ or $\mathbf C = \mathbf C ^ {1}$ can consist of any number $k$ of connected components, $0 \leq k \leq \infty$. If $D$ is regarded as a domain in the compact extended plane $\overline{\mathbf R}\; {} ^ {2}$ or $\overline{\mathbf C}\;$ and the number $k$ of boundary components is finite, then $k$ is called the connectivity order of $D$; for $k > 1$, $D$ is called multiply connected. In other words, the connectivity order $k$ is one more than the minimum number of cross-cuts joining components of the boundary in pairs that are necessary to make $D$ simply connected. For $k = 2$, $D$ is called doubly connected, for $k = 3$, triply connected, etc.; for $k < \infty$ one has finitely-connected domains and for $k = \infty$ infinitely-connected domains. The connectivity order of a plane domain characterizes its topological type. The topological types of domains in $\mathbf R ^ {n}$, $n \geq 3$, or in $\mathbf C ^ {m}$, $m \geq 2$, cannot be characterized by a single number.

Even for a simply-connected plane domain $D$ the metric structure of the boundary $\textrm{ Fr } D$ can be very complicated (see Limit elements). In particular, the boundary points can be divided into accessible points $x _ {0} \in \textrm{ Fr } D$, for which there exists a path $x ( t)$, $0 \leq t \leq 1$, $x ( 0) \in D$, $x ( 1) = x _ {0}$, joining $x _ {0}$ in $D$ with any point $x ( 0) \in D$, and inaccessible points, for which no such paths exists (cf. Attainable boundary point). For any simply-connected plane domain $D$ the set of accessible points of $\textrm{ Fr } D$ is everywhere dense in $\textrm{ Fr } D$.

A domain $D$ in $\mathbf R ^ {n}$ or $\mathbf C ^ {m}$ is called bounded, or finite, if

$$\sup \ \{ {| x | } : { x \in D } \} < \infty ;$$

if not, $D$ is called unbounded or infinite. A closed plane Jordan curve divides the plane $\mathbf R ^ {2}$ or $\mathbf C$ into two Jordan domains: A finite domain $D ^ {+}$ and an infinite domain $D ^ {-}$. All boundary points of a Jordan domain are accessible.

Instead of $\textrm{ Fr } D$, the boundary of $D$ is also denoted by $\textrm{ b } D$ or $\partial D$.