Domain

A non-empty connected open set in a topological space . The closure of a domain is called a closed domain; the closed set is called the boundary of . The points are also called the interior points of ; the points are called the boundary points of ; the points of the complement are called the exterior points of .

Any two points of a domain in the real Euclidean space , (or in the complex space , , or on a Riemann surface or in a Riemannian domain), can be joined by a path (or arc) lying completely in ; if or , 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 ; their boundaries consist of at most two points. A domain in the plane is called simply connected if any closed path in can be continuously deformed to a point, remaining throughout in . In general, the boundary of a simply-connected domain in the (open) plane or can consist of any number of connected components, . If is regarded as a domain in the compact extended plane or and the number of boundary components is finite, then is called the connectivity order of ; for , is called multiply connected. In other words, the connectivity order is one more than the minimum number of cross-cuts joining components of the boundary in pairs that are necessary to make simply connected. For , is called doubly connected, for , triply connected, etc.; for one has finitely-connected domains and for infinitely-connected domains. The connectivity order of a plane domain characterizes its topological type. The topological types of domains in , , or in , , cannot be characterized by a single number.

Even for a simply-connected plane domain the metric structure of the boundary can be very complicated (see Limit elements). In particular, the boundary points can be divided into accessible points , for which there exists a path , , , , joining in with any point , and inaccessible points, for which no such paths exists (cf. Attainable boundary point). For any simply-connected plane domain the set of accessible points of is everywhere dense in .

A domain in or is called bounded, or finite, if

if not, is called unbounded or infinite. A closed plane Jordan curve divides the plane or into two Jordan domains: A finite domain and an infinite domain . All boundary points of a Jordan domain are accessible.

Comments

Instead of , the boundary of is also denoted by or .

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.