Domain

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

Any two points of a domain $D$ in the real Euclidean space ${\mathbf{R}}^n$, $n\ge 1$( or in the complex space ${\mathbf{C}}^m$, $m\ge 1$, 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, $0\le k\le \mathrm{\infty }$. If $D$ is regarded as a domain in the compact extended plane $\bar{\mathbf{R}}{}^2$ or $\bar{\mathbf{C}}$ 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 $k<\mathrm{\infty }$ one has finitely-connected domains and for $k=\mathrm{\infty }$ infinitely-connected domains. The connectivity order of a plane domain characterizes its topological type. The topological types of domains in ${\mathbf{R}}^n$, $n\ge 3$, or in ${\mathbf{C}}^m$, $m\ge 2$, cannot be characterized by a single number.

Even for a simply-connected plane domain $D$ the metric structure of the boundary $\text{ Fr }D$ can be very complicated (see Limit elements). In particular, the boundary points can be divided into accessible points $x_0\in \text{ Fr }D$, for which there exists a path $x(t)$, $0\le t\le 1$, $x(0)\in D$, $x(1)=x_0$, 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 $\text{ Fr }D$ is everywhere dense in $\text{ Fr }D$.

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

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

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.

Comments[edit]

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

From the definition it can be seen that a domain is bounded if (and only if) it is contained in a ball centred at the coordinate origin and of finite radius.