Domain (mathematical analysis)
{{short description|Connected open subset of a topological space}}
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 {{math|Rn}} or the complex coordinate space {{math|Cn}}. A connected open subset of coordinate space is frequently used for the domain of a function.Be it noted that, nevertheless, functions may be defined on sets that are not topological spaces: for more details, consult the relevant wikipedia entry.
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,For instance {{Harv|Sveshnikov|Tikhonov|1978|loc=[https://archive.org/details/SveshnikovTikhonovTheTheoryOfFunctionsOfAComplexVariable/page/n24/ §1.3 pp. 21–22]}}. some use the term region,For instance {{Harv|Churchill|1948|loc=[https://archive.org/details/introductiontoco00chur/page/16 §1.9 pp. 16–17]}}; {{Harv|Ahlfors|1953|loc=[https://archive.org/details/complexanalysisi00ahlf/page/58/ §2.2 p. 58]}}; {{Harv|Rudin|1974|loc=[https://archive.org/details/realcomplexanaly00rudi_0/page/213/ §10.1 p. 213]}} reserves the term domain for the domain of a function; {{Harv|Carathéodory|1964|loc=[https://archive.org/details/theoryoffunction0001cara/page/97/ p. 97]}} uses the term region for a connected open set and the term continuum for a connected closed set. some use both terms interchangeably,For instance {{Harv|Townsend|1915|loc=[https://archive.org/details/functionsofcompl00towniala/page/20/ §10, p. 20]}}; {{Harv|Carrier|Krook|Pearson|1966|loc=[https://archive.org/details/functionsofcompl00carr/page/32/ §2.2 p. 32]}}. and some define the two terms slightly differently;For instance {{Harv|Churchill|1960|loc=[https://archive.org/details/isbn_9780070108530/page/17/ §1.9 p. 17]}}, who does not require that a region be connected or open. some avoid ambiguity by sticking with a phrase such as non-empty connected open subset.For instance {{Harv|Dieudonné|1960|loc=[https://archive.org/details/foundationsofmod00dieu/page/64/ §3.19 pp. 64–67]}} generally uses the phrase open connected set, but later defines simply connected domain ([https://archive.org/details/foundationsofmod00dieu/page/215 §9.7 p. 215]); {{cite web |last=Tao |first=Terence |author-link= Terence Tao |date=2016 |title=246A, Notes 2: complex integration |url=https://terrytao.wordpress.com/2016/09/27/246a-notes-2-complex-integration/}}, also, {{harv|Bremermann|1956}} called the region an open set and the domain a concatenated open set.
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.For instance {{Harv|Fuchs|Shabat|1964|loc=[https://archive.org/details/functionsofcompl0001fuks/page/22/ §6 pp. 22–23]}};
{{Harv|Kreyszig|1972|loc=[https://archive.org/details/advancedengineer00krey/page/469/ §11.1 p. 469]}}; {{Harv|Kwok|2002|loc=§1.4, p. 23.}} 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, Smooth function 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 {{math|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 {{math|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
{{quote
|text=
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.
{{langx|de|Eine offene Punktmenge heißt zusammenhängend, wenn man sie nicht als Summe von zwei offenen Punktmengen darstellen kann. Eine offene zusammenhängende Punktmenge heißt ein Gebiet.}}
|multiline=yes
|sign= Constantin Carathéodory
|source={{Harv|Carathéodory|1918|p=222}}
}}
According to Hans Hahn,See {{harv|Hahn|1921|loc=[https://archive.org/details/theoriederreelle01hahnuoft/page/85 p. 85 footnote 1]}}. the concept of a domain as an open connected set was introduced by Constantin Carathéodory in his famous book {{harv|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.{{harvtxt|Hahn|1921|loc=[https://archive.org/details/theoriederreelle01hahnuoft/page/61/ 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." 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.For example {{Harv|Forsyth|1893}} uses the term region informally throughout (e.g. [https://archive.org/details/theoryoffunction00fors/page/21/ §16, p. 21]) alongside the informal expression part of the {{mvar|z}}-plane, and defines the domain of a point {{mvar|a}} for a function {{mvar|f}} to be the largest neighbourhood (mathematics)#In a metric space of {{mvar|a}} in which {{mvar|f}} is holomorphic ([https://archive.org/details/theoryoffunction00fors/page/52/ §32, p. 52]). The first edition of the influential textbook {{Harv|Whittaker|1902}} uses the terms domain and region informally and apparently interchangeably. By the second edition {{Harv|Whittaker|Watson|1915|loc=[https://archive.org/details/courseofmodernan00whituoft/page/44/ §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. {{Harv|Goursat|1905|loc=[https://archive.org/details/courseinmathemat02gouruoft/page/10/ §262, p. 10]}} defines région [region] or aire [area] as a connected portion of the plane. {{Harv|Townsend|1915|loc=[https://archive.org/details/functionsofcompl00towniala/page/20/ §10, p. 20]}} defines a region or domain to be a connected portion of the complex plane consisting only of inner points.
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,See {{harvs|last= Miranda|year1=1955|year2=1970|loc1=p. 1|loc2=p. 2}}.Precisely, in the first edition of his monograph, {{harvtxt|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". and reserves the term "domain" to identify an internally connected,An internally connected set is a set whose interior is connected. perfect set, each point of which is an accumulation point of interior points, following his former master Mauro Picone:See {{harv|Picone|1923|p=66}}. according to this convention, if a set {{math|A}} is a region then its closure {{math|{{overline|A}}}} is a domain.
See also
- {{annotated link|Analytic polyhedron}}
- {{annotated link|Caccioppoli set}}
- {{annotated link|Hermitian symmetric space#Classical domains}}
- {{annotated link|Interval (mathematics)}}
- {{annotated link|Lipschitz domain}}
- {{annotated link|Whitehead's point-free geometry}}
Notes
{{reflist|29em}}
References
- {{cite book |last=Ahlfors |first=Lars |author-link=Lars Ahlfors |date=1953 |title=Complex Analysis |publisher=McGraw-Hill |url=https://archive.org/details/complexanalysisi00ahlf/ |url-access=limited }}
- {{cite journal |jstor=1992976|title=Complex Convexity|last1=Bremermann|first1=H. J.|journal=Transactions of the American Mathematical Society|year=1956|volume=82|issue=1|pages=17–51|doi=10.1090/S0002-9947-1956-0079100-2|doi-access=free}}
- {{cite book |last=Carathéodory |first=Constantin |author-link=Constantin Carathéodory |date=1918 |title=Vorlesungen über reelle Funktionen |trans-title=Lectures on real functions |publisher=B. G. Teubner |language=de |mr=0225940 |jfm=46.0376.12 }} Reprinted 1968 (Chelsea).
- {{cite book |last=Carathéodory |first=Constantin |date=1964 |orig-date=1954 |edition=2nd |title=Theory of Functions of a Complex Variable, vol. I |publisher=Chelsea |url=https://archive.org/details/theoryoffunction0001cara/ |url-access=limited }} English translation of {{cite book |last=Carathéodory |first=Constantin |date=1950 |title=Functionentheorie I |publisher=Birkhäuser |lang=de }}
- {{cite book |last1=Carrier |first1=George |author-link1=George F. Carrier |last2=Krook |first2=Max |author-link2=Max Krook |last3=Pearson |first3=Carl |date=1966 |title=Functions of a Complex Variable: Theory and Technique |publisher=McGraw-Hill |url=https://archive.org/details/functionsofcompl00carr/ |url-access=limited }}
- {{cite book |last=Churchill |first=Ruel |author-link=Ruel Vance Churchill |date=1948 |title=Introduction to Complex Variables and Applications |edition=1st |publisher=McGraw-Hill |url=https://archive.org/details/introductiontoco00chur/ |url-access=limited }}
{{cite book |last=Churchill |first=Ruel |date=1960 |title=Complex Variables and Applications |edition=2nd |publisher=McGraw-Hill |isbn=9780070108530 |url=https://archive.org/details/isbn_9780070108530/ |url-access=limited }} - {{cite book |last=Dieudonné |first=Jean |author-link=Jean Dieudonné |date=1960 |title=Foundations of Modern Analysis |publisher=Academic Press |url=https://archive.org/details/foundationsofmod00dieu/ |url-access=limited }}
- {{cite book |last=Eves |first=Howard |author-link=Howard Eves |date=1966 |title=Functions of a Complex Variable |page=105 |publisher=Prindle, Weber & Schmidt }}
- {{cite book |last=Forsyth |first=Andrew |author-link=Andrew Forsyth |title=Theory of Functions of a Complex Variable |date=1893 |url=https://archive.org/details/theoryoffunction00fors/?q=region |publisher=Cambridge |jfm=25.0652.01 }}
- {{cite book |last1=Fuchs |first1=Boris |last2=Shabat |first2=Boris |date=1964 |title=Functions of a complex variable and some of their applications, vol. 1 |publisher=Pergamon |url=https://archive.org/details/functionsofcompl0001fuks/ |url-access=limited }} English translation of {{cite book |last1=Фукс |first1=Борис |last2=Шабат |first2=Борис |date=1949 |title=Функции комплексного переменного и некоторые их приложения |lang=ru |publisher=Физматгиз |url=https://ikfia.ysn.ru/wp-content/uploads/2018/01/FuksShabat1964ru.pdf }}
- {{cite book |author-link=Édouard Goursat |last=Goursat |first=Édouard |date=1905 |title=Cours d'analyse mathématique, tome 2 |lang=fr |trans-title=A course in mathematical analysis, vol. 2 |publisher=Gauthier-Villars |url=https://archive.org/details/courseinmathemat02gouruoft/ }}
- {{cite book |last=Hahn |first=Hans |author-link=Hans Hahn (mathematician) |title=Theorie der reellen Funktionen. Erster Band |trans-title=Theory of Real Functions, vol. I |publisher=Springer |year=1921 |language=de |url=https://archive.org/details/theoriederreelle01hahnuoft/ |jfm=48.0261.09 }}
- {{cite book |last1=Krantz |first1=Steven |author-link1=Steven G. Krantz |last2=Parks |first2=Harold |author-link2=Harold R. Parks |date=1999 |title=The Geometry of Domains in Space |publisher=Birkhäuser }}
- {{cite book |last=Kreyszig |first=Erwin |author-link=Erwin Kreyszig |date=1972 |title=Advanced Engineering Mathematics |orig-date=1962 |edition=3rd |publisher=Wiley |isbn=9780471507284 |url=https://archive.org/details/advancedengineer00krey/ |url-access=limited }}
- {{cite book |last=Kwok |first=Yue-Kuen |date=2002 |title=Applied Complex Variables for Scientists and Engineers |publisher=Cambridge }}
- {{cite book |last=Miranda |first=Carlo |author-link=Carlo Miranda |title=Equazioni alle derivate parziali di tipo ellittico |lang=it |publisher=Springer |year=1955 |mr=0087853 |zbl=0065.08503 }} Translated as {{cite book |last=Miranda |first=Carlo |author-link=Carlo Miranda |title=Partial Differential Equations of Elliptic Type |date=1970 |publisher=Springer |translator-last1=Motteler |translator-first1=Zane C. |edition=2nd |mr=0284700 |zbl=0198.14101 }}
- {{cite book |last=Picone |first=Mauro |author-link=Mauro Picone |date=1923 |title=Lezioni di analisi infinitesimale, vol. I |trans-title=Lessons in infinitesimal analysis |publisher=Circolo matematico di Catania |chapter=Parte Prima – La Derivazione |language=it |chapter-url=https://matematicaitaliana.sns.it/media/volumi/462/picone_parte_I.pdf |jfm=49.0172.07 }}
- {{cite book |last=Rudin |first=Walter |author-link=Walter Rudin |date=1974 |orig-date=1966 |title=Real and Complex Analysis |edition=2nd |publisher=McGraw-Hill |isbn=9780070542334 |url=https://archive.org/details/realcomplexanaly00rudi_0/ |url-access=limited }}
- {{Eom |title=Domain |author-last1=Solomentsev |author-first1=Evgeny |oldid=46762 }}
- {{cite book| last1=Sveshnikov |first1=Aleksei |author-link1=Aleksei Sveshnikov |last2=Tikhonov |first2=Andrey |author-link2=Andrey Nikolayevich Tikhonov |date=1978 |title=The Theory Of Functions Of A Complex Variable |publisher=Mir |url=https://archive.org/details/SveshnikovTikhonovTheTheoryOfFunctionsOfAComplexVariable/}} English translation of {{cite book |last1=Свешников |first1=Алексей |last2=Ти́хонов |first2=Андре́й |date=1967 |title=Теория функций комплексной переменной |lang=ru |publisher=Наука }}
- {{cite book |last=Townsend |first=Edgar |title=Functions of a Complex Variable |date=1915 |publisher=Holt |url=https://archive.org/details/functionsofcompl00towniala/?q=region }}
- {{cite book |title=A Course Of Modern Analysis |last=Whittaker |first=Edmund |author-link=Edmund Taylor Whittaker |date=1902 |publisher=Cambridge |edition=1st |url=https://archive.org/details/courseofmoder1st00whituoft/?q=region |jfm=33.0390.01}}
{{cite book |title=A Course Of Modern Analysis |last1=Whittaker |first1=Edmund |last2=Watson |first2=George |author-link2=George Neville Watson |date=1915 |publisher=Cambridge |edition=2nd |url=https://archive.org/details/courseofmodernan00whituoft/?q=region }}
Category:Mathematical analysis