Conventions[edit]
One common convention is to define a domain as a connected open set but a region as the union of a domain with none, some, or all of its limit points.[6] A closed region or closed domain is the union of a domain and all of its limit points.
Various degrees of smoothness of the boundary of the domain are required for various properties of functions defined on the domain to hold, such as integral theorems (Green's theorem, Stokes theorem), properties of Sobolev spaces, and to define measures on the boundary and spaces of traces (generalized functions defined on the boundary). Commonly considered types of domains are domains with continuous boundary, Lipschitz boundary, C1 boundary, and so forth.
A bounded domain is a domain that is bounded, i.e., contained in some ball. Bounded region is defined similarly. An exterior domain or external domain is a domain whose complement is bounded; sometimes smoothness conditions are imposed on its boundary.
In complex analysis, a complex domain (or simply domain) is any connected open subset of the complex plane C. For example, the entire complex plane is a domain, as is the open unit disk, the open upper half-plane, and so forth. Often, a complex domain serves as the domain of definition for a holomorphic function. In the study of several complex variables, the definition of a domain is extended to include any connected open subset of Cn.
In Euclidean spaces, the extent of one-, two-, and three-dimensional regions are called, respectively, length, area, and volume.