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 Rn or the complex coordinate space Cn. A connected open subset of coordinate space is frequently used for the domain of a function.
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, some use the term region, some use both terms interchangeably, and some define the two terms slightly differently; some avoid ambiguity by sticking with a phrase such as non-empty connected open subset.
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. 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, 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, 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. 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.
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, and reserves the term "domain" to identify an internally connected,perfect set, each point of which is an accumulation point of interior points, following his former master Mauro Picone: according to this convention, if a set A is a region then its closure A is a domain.
See also
- Analytic polyhedron – Subset of complex n-space bounded by analytic functions
- Caccioppoli set – Region with boundary of finite measure
- Hermitian symmetric space#Classical domains – Manifold with inversion symmetry
- Interval (mathematics) – All numbers between two given numbers
- Lipschitz domain
- Whitehead's point-free geometry – Geometric theory based on regions
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.
- 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).
References
- Ahlfors, Lars (1953). Complex Analysis. McGraw-Hill.
- Bremermann, H. J. (1956). "Complex Convexity". Transactions of the American Mathematical Society. 82 (1): 17–51. doi:10.1090/S0002-9947-1956-0079100-2. JSTOR 1992976.
- Carathéodory, Constantin (1918). Vorlesungen über reelle Funktionen [Lectures on real functions] (in German). B. G. Teubner. JFM 46.0376.12. MR 0225940. Reprinted 1968 (Chelsea).
- Carathéodory, Constantin (1964) [1954]. Theory of Functions of a Complex Variable, vol. I (2nd ed.). Chelsea. English translation of Carathéodory, Constantin (1950). Functionentheorie I (in German). Birkhäuser.
- Carrier, George; Krook, Max; Pearson, Carl (1966). Functions of a Complex Variable: Theory and Technique. McGraw-Hill.
- Churchill, Ruel (1948). Introduction to Complex Variables and Applications (1st ed.). McGraw-Hill.
Churchill, Ruel (1960). Complex Variables and Applications (2nd ed.). McGraw-Hill. ISBN 9780070108530. - Dieudonné, Jean (1960). Foundations of Modern Analysis. Academic Press.
- Eves, Howard (1966). Functions of a Complex Variable. Prindle, Weber & Schmidt. p. 105.
- Forsyth, Andrew (1893). Theory of Functions of a Complex Variable. Cambridge. JFM 25.0652.01.
- Fuchs, Boris; Shabat, Boris (1964). Functions of a complex variable and some of their applications, vol. 1. Pergamon. English translation of Фукс, Борис; Шабат, Борис (1949). Функции комплексного переменного и некоторые их приложения (PDF) (in Russian). Физматгиз.
- Goursat, Édouard (1905). Cours d'analyse mathématique, tome 2 [A course in mathematical analysis, vol. 2] (in French). Gauthier-Villars.
- Hahn, Hans (1921). Theorie der reellen Funktionen. Erster Band [Theory of Real Functions, vol. I] (in German). Springer. JFM 48.0261.09.
- Krantz, Steven; Parks, Harold (1999). The Geometry of Domains in Space. Birkhäuser.
- Kreyszig, Erwin (1972) [1962]. Advanced Engineering Mathematics (3rd ed.). Wiley. ISBN 9780471507284.
- Kwok, Yue-Kuen (2002). Applied Complex Variables for Scientists and Engineers. Cambridge.
- Miranda, Carlo (1955). Equazioni alle derivate parziali di tipo ellittico (in Italian). Springer. MR 0087853. Zbl 0065.08503. Translated as Miranda, Carlo (1970). Partial Differential Equations of Elliptic Type. Translated by Motteler, Zane C. (2nd ed.). Springer. MR 0284700. Zbl 0198.14101.
- Picone, Mauro (1923). "Parte Prima – La Derivazione" (PDF). Lezioni di analisi infinitesimale, vol. I [Lessons in infinitesimal analysis] (in Italian). Circolo matematico di Catania. JFM 49.0172.07.
- Rudin, Walter (1974) [1966]. Real and Complex Analysis (2nd ed.). McGraw-Hill. ISBN 9780070542334.
- Solomentsev, Evgeny (2001) [1994], "Domain", Encyclopedia of Mathematics, EMS Press
- Sveshnikov, Aleksei; Tikhonov, Andrey (1978). The Theory Of Functions Of A Complex Variable. Mir. English translation of Свешников, Алексей; Ти́хонов, Андре́й (1967). Теория функций комплексной переменной (in Russian). Наука.
- Townsend, Edgar (1915). Functions of a Complex Variable. Holt.
- Whittaker, Edmund (1902). A Course Of Modern Analysis (1st ed.). Cambridge. JFM 33.0390.01.
Whittaker, Edmund; Watson, George (1915). A Course Of Modern Analysis (2nd ed.). Cambridge.
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 Rn or the complex coordinate space Cn A connected open subset of coordinate space is frequently used for the domain of a function 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 some use the term region some use both terms interchangeably and some define the two terms slightly differently some avoid ambiguity by sticking with a phrase such as non empty connected open subset ConventionsOne 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 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 one two and three dimensional regions are curves surfaces and solids whose extent are called respectively length area and volume Historical notesDefinition 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 zusammenhangend wenn man sie nicht als Summe von zwei offenen Punktmengen darstellen kann Eine offene zusammenhangende Punktmenge heisst ein Gebiet Constantin Caratheodory Caratheodory 1918 p 222 According to Hans Hahn the concept of a domain as an open connected set was introduced by Constantin Caratheodory in his famous book Caratheodory 1918 In this definition Caratheodory considers obviously non empty disjoint sets Hahn also remarks that the word Gebiet Domain was occasionally previously used as a synonym of open set 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 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 and reserves the term domain to identify an internally connected perfect set each point of which is an accumulation point of interior points following his former master Mauro Picone according to this convention if a set A is a region then its closure A is a domain See alsoAnalytic polyhedron Subset of complex n space bounded by analytic functions Caccioppoli set Region with boundary of finite measure Hermitian symmetric space Classical domains Manifold with inversion symmetry Interval mathematics All numbers between two given numbers Lipschitz domain Whitehead s point free geometry Geometric theory based on regionsNotesHowever functions may be defined on sets that are not topological spaces For instance Sveshnikov amp 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 Caratheodory 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 amp 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 Dieudonne 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 amp 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 fur 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 amp 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 region 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 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 ReferencesAhlfors Lars 1953 Complex Analysis McGraw Hill Bremermann H J 1956 Complex Convexity Transactions of the American Mathematical Society 82 1 17 51 doi 10 1090 S0002 9947 1956 0079100 2 JSTOR 1992976 Caratheodory Constantin 1918 Vorlesungen uber reelle Funktionen Lectures on real functions in German B G Teubner JFM 46 0376 12 MR 0225940 Reprinted 1968 Chelsea Caratheodory Constantin 1964 1954 Theory of Functions of a Complex Variable vol I 2nd ed Chelsea English translation of Caratheodory Constantin 1950 Functionentheorie I in German Birkhauser Carrier George Krook Max Pearson Carl 1966 Functions of a Complex Variable Theory and Technique McGraw Hill Churchill Ruel 1948 Introduction to Complex Variables and Applications 1st ed McGraw Hill Churchill Ruel 1960 Complex Variables and Applications 2nd ed McGraw Hill ISBN 9780070108530 Dieudonne Jean 1960 Foundations of Modern Analysis Academic Press Eves Howard 1966 Functions of a Complex Variable Prindle Weber amp Schmidt p 105 Forsyth Andrew 1893 Theory of Functions of a Complex Variable Cambridge JFM 25 0652 01 Fuchs Boris Shabat Boris 1964 Functions of a complex variable and some of their applications vol 1 Pergamon English translation of Fuks Boris Shabat Boris 1949 Funkcii kompleksnogo peremennogo i nekotorye ih prilozheniya PDF in Russian Fizmatgiz Goursat Edouard 1905 Cours d analyse mathematique tome 2 A course in mathematical analysis vol 2 in French Gauthier Villars Hahn Hans 1921 Theorie der reellen Funktionen Erster Band Theory of Real Functions vol I in German Springer JFM 48 0261 09 Krantz Steven Parks Harold 1999 The Geometry of Domains in Space Birkhauser Kreyszig Erwin 1972 1962 Advanced Engineering Mathematics 3rd ed Wiley ISBN 9780471507284 Kwok Yue Kuen 2002 Applied Complex Variables for Scientists and Engineers Cambridge Miranda Carlo 1955 Equazioni alle derivate parziali di tipo ellittico in Italian Springer MR 0087853 Zbl 0065 08503 Translated as Miranda Carlo 1970 Partial Differential Equations of Elliptic Type Translated by Motteler Zane C 2nd ed Springer MR 0284700 Zbl 0198 14101 Picone Mauro 1923 Parte Prima La Derivazione PDF Lezioni di analisi infinitesimale vol I Lessons in infinitesimal analysis in Italian Circolo matematico di Catania JFM 49 0172 07 Rudin Walter 1974 1966 Real and Complex Analysis 2nd ed McGraw Hill ISBN 9780070542334 Solomentsev Evgeny 2001 1994 Domain Encyclopedia of Mathematics EMS Press Sveshnikov Aleksei Tikhonov Andrey 1978 The Theory Of Functions Of A Complex Variable Mir English translation of Sveshnikov Aleksej Ti honov Andre j 1967 Teoriya funkcij kompleksnoj peremennoj in Russian Nauka Townsend Edgar 1915 Functions of a Complex Variable Holt Whittaker Edmund 1902 A Course Of Modern Analysis 1st ed Cambridge JFM 33 0390 01 Whittaker Edmund Watson George 1915 A Course Of Modern Analysis 2nd ed Cambridge
