Region (mathematics)
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Region Mathematics

In mathematical analysis, a domain or region is a non-empty connected open set in a topological space, in particular any non-empty connected open subset of the real coordinate space Rn or the complex coordinate space Cn. This is a different concept than the domain of a function, though it is often used for that purpose, for example in partial differential equations and Sobolev spaces.

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 which is a bounded set, while an exterior or external domain is the interior of the complement of a bounded domain.

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.

## Historical notes

Definition. Eine offene Punktmenge heißt zusammenhängend, wenn man sie nicht als Summe von zwei offenen Punktmengen darstellen kann. Eine offene zusammenhängende Punktmenge heißt ein Gebiet.[1]

According to Hans Hahn,[2] the concept of a domain as an open connected set was introduced by Constantin Carathéodory in his famous book (Carathéodory 1918). Hahn also remarks that the word "Gebiet" ("Domain") was occasionally previously used as a synonym of open set.[3] 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.[4]

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,[5][6] and reserves the term "domain" to identify an internally connected,[7] perfect set, each point of which is an accumulation point of interior points,[5] following his former master Mauro Picone:[8] according to this convention, if a set A is a region then its closure A is a domain.[5]

According to Kreyszig,

A region is a set consisting of a domain plus, perhaps, some or all of its boundary points. (The reader is warned that some authors use the term "region" for what we call a domain [following standard terminology], and others make no distinction between the two terms.)[9]

According to Yue Kuen Kwok,

An open connected set is called an open region or domain. ...to an open region we may add none, some, or all its limit points, and simply call the new set a region.[10]

## Notes

1. ^ English: "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": in this definition, Carathéodory considers obviously non-empty disjoint sets.
2. ^ See (Hahn 1921, p. 85 footnote 1).
3. ^ 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."
4. ^ 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. (Townsend 1915, §10, p. 20) defines a region or domain to be a connected portion of the complex plane consisting only of inner points.
5. ^ a b c See (Miranda 1955, p. 1, 1970, p. 2).
6. ^ 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".
7. ^ An internally connected set is a set whose interior is connected.
8. ^ See (Picone 1923, p. 66).
9. ^ Erwin Kreyszig (1993) Advanced Engineering Mathematics, 7th edition, p. 720, John Wiley & Sons, ISBN 0-471-55380-8
10. ^ Yue Kuen Kwok (2002) Applied Complex Variables for Scientists and Engineers, § 1.4 Some topological definitions, p 23, Cambridge University Press, ISBN 0-521-00462-4