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.

Comments

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