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In mathematical analysis, a domain or region is a non-empty, connected, and open set in a topological space. In particular, it is any non-empty connected open subset of the real coordinate space $R n$ or the complex coordinate space $C n$ . A connected open subset of coordinate space is frequently used for the domain of a function.^[a]

The basic idea of a connected subset of a space dates from the 19th century, but precise definitions vary slightly from generation to generation, author to author, and edition to edition, as concepts developed and terms were translated between German, French, and English works. In English, some authors use the term domain,^[1] some use the term region,^[2] some use both terms interchangeably,^[3] and some define the two terms slightly differently;^[4] some avoid ambiguity by sticking with a phrase such as non-empty connected open subset.^[5]

Conventions

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, $C 1$ 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 $C n$ .

In Euclidean spaces, one-, two-, and three-dimensional regions are curves, surfaces, and solids, whose extent are called, respectively, length, area, and volume.

Historical notes

Definition. An open set is connected if it cannot be expressed as the sum of two open sets. An open connected set is called a domain.
German: Eine offene Punktmenge heißt zusammenhängend, wenn man sie nicht als Summe von zwei offenen Punktmengen darstellen kann. Eine offene zusammenhängende Punktmenge heißt ein Gebiet.

— Constantin Carathéodory, (Carathéodory 1918, p. 222)

According to Hans Hahn,^[7] the concept of a domain as an open connected set was introduced by Constantin Carathéodory in his famous book (Carathéodory 1918). In this definition, Carathéodory considers obviously non-empty disjoint sets. Hahn also remarks that the word "Gebiet" ("Domain") was occasionally previously used as a synonym of open set.^[8] The rough concept is older. In the 19th and early 20th century, the terms domain and region were often used informally (sometimes interchangeably) without explicit definition.^[9]

However, the term "domain" was occasionally used to identify closely related but slightly different concepts. For example, in his influential monographs on elliptic partial differential equations, Carlo Miranda uses the term "region" to identify an open connected set,^[10]^[11] and reserves the term "domain" to identify an internally connected,^[12] perfect set, each point of which is an accumulation point of interior points,^[10] following his former master Mauro Picone:^[13] according to this convention, if a set $A$ is a region then its closure $A$ is a domain.^[10]

Notes

^ However, functions may be defined on sets that are not topological spaces.

^ For instance (Sveshnikov & Tikhonov 1978, §1.3 pp. 21–22).
^ For instance (Churchill 1948, §1.9 pp. 16–17); (Ahlfors 1953, §2.2 p. 58); (Rudin 1974, §10.1 p. 213) reserves the term domain for the domain of a function; (Carathéodory 1964, p. 97) uses the term region for a connected open set and the term continuum for a connected closed set.
^ For instance (Townsend 1915, §10, p. 20); (Carrier, Krook & Pearson 1966, §2.2 p. 32).
^ For instance (Churchill 1960, §1.9 p. 17), who does not require that a region be connected or open.
^ For instance (Dieudonné 1960, §3.19 pp. 64–67) generally uses the phrase open connected set, but later defines simply connected domain (§9.7 p. 215); Tao, Terence (2016). "246A, Notes 2: complex integration"., also, (Bremermann 1956) called the region an open set and the domain a concatenated open set.
^ For instance (Fuchs & Shabat 1964, §6 pp. 22–23); (Kreyszig 1972, §11.1 p. 469); (Kwok 2002, §1.4, p. 23.)
^ See (Hahn 1921, p. 85 footnote 1).
^ Hahn (1921, p. 61 footnote 3), commenting the just given definition of open set ("offene Menge"), precisely states:-"Vorher war, für diese Punktmengen die Bezeichnung "Gebiet" in Gebrauch, die wir (§ 5, S. 85) anders verwenden werden." (Free English translation:-"Previously, the term "Gebiet" was occasionally used for such point sets, and it will be used by us in (§ 5, p. 85) with a different meaning."
^ For example (Forsyth 1893) uses the term region informally throughout (e.g. §16, p. 21) alongside the informal expression part of the $z$ -plane, and defines the domain of a point $a$ for a function $f$ to be the largest $r$ -neighborhood of $a$ in which $f$ is holomorphic (§32, p. 52). The first edition of the influential textbook (Whittaker 1902) uses the terms domain and region informally and apparently interchangeably. By the second edition (Whittaker & Watson 1915, §3.21, p. 44) define an open region to be the interior of a simple closed curve, and a closed region or domain to be the open region along with its boundary curve. (Goursat 1905, §262, p. 10) defines région [region] or aire [area] as a connected portion of the plane. (Townsend 1915, §10, p. 20) defines a region or domain to be a connected portion of the complex plane consisting only of inner points.
^ ^a ^b ^c See (Miranda 1955, p. 1, 1970, p. 2).
^ Precisely, in the first edition of his monograph, Miranda (1955, p. 1) uses the Italian term "campo", meaning literally "field" in a way similar to its meaning in agriculture: in the second edition of the book, Zane C. Motteler appropriately translates this term as "region".
^ An internally connected set is a set whose interior is connected.
^ See (Picone 1923, p. 66).