View source for Domain (mathematical analysis)

{{short description|Connected open subset of a topological space}}

In [[mathematical analysis]], a '''domain''' or '''region''' is a [[empty set|non-empty]], [[connected (topology)|connected]], and [[open set]] in a [[topological space]]. In particular, it is any non-empty connected open [[subset]] of the [[real coordinate space]] {{math|'''R'''<sup>''n''</sup>}} or the [[complex coordinate space]] {{math|'''C'''<sup>''n''</sup>}}. A connected open subset of [[coordinate space]] is frequently used for the [[domain of a function]].{{efn|However, [[function (mathematics)|function]]s may be defined on sets that are not topological spaces.}}

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'',<ref>For instance {{Harv|Sveshnikov|Tikhonov|1978|loc=[https://archive.org/details/SveshnikovTikhonovTheTheoryOfFunctionsOfAComplexVariable/page/n24/ §1.3 pp. 21–22]}}.</ref> some use the term ''region'',<ref>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.</ref> some use both terms interchangeably,<ref>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]}}.</ref> and some define the two terms slightly differently;<ref>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.</ref> some avoid ambiguity by sticking with a phrase such as ''non-empty connected open subset''.<ref>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 space|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.</ref>

==Conventions==
One common convention is to define a ''domain'' as a connected open set but a ''region'' as the [[union (set theory)|union]] of a domain with none, some, or all of its [[limit point]]s.<ref>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.}}</ref> 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 (mathematics)|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 space]]s, and to define [[Measure (mathematics)|measures]] on the boundary and spaces of [[Trace operator|traces]] (generalized functions defined on the boundary). Commonly considered types of domains are domains with [[continuous function|continuous]] boundary, [[Lipschitz domain|Lipschitz boundary]], [[Smooth function|{{math|''C''<sup>1</sup>}}]] boundary, and so forth.

A '''bounded domain''' is a domain that is [[bounded set|bounded]], i.e., contained in some ball.  '''Bounded region''' is defined similarly.  An '''exterior domain''' or '''external domain''' is a domain whose [[Complement (set theory)|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 a function|domain of definition]] for a [[holomorphic function]]. In the study of [[Function of several complex variables|several complex variables]], the definition of a domain is extended to include any connected open subset of {{math|'''C'''<sup>''n''</sup>}}.

In [[Euclidean space]]s, [[One-dimensional|one-]], [[Two-dimensional|two-]], and [[three-dimensional]] regions are [[curve]]s, [[Surface (mathematics)|surface]]s, and [[Solid (mathematics)|solids]], whose extent are called, respectively, ''[[arc length|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 (mathematician)|Hans Hahn]],<ref>See {{harv|Hahn|1921|loc=[https://archive.org/details/theoriederreelle01hahnuoft/page/85 p. 85 footnote 1]}}.</ref> 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 [[empty set|non-empty]] [[disjoint sets|disjoint]] sets.
Hahn also remarks that the word "''Gebiet''" ("''Domain''") was occasionally previously used as a [[synonym]] of [[open set]].<ref>{{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.''"</ref>  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.<ref>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|{{mvar|r}}-neighborhood]] of {{mvar|a}} in which {{mvar|f}} is [[holomorphic function|holomorphic]] ([https://archive.org/details/theoryoffunction00fors/page/52/ §32, p. 52]). The first edition of the [[A Course of Modern Analysis|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 [[Jordan curve theorem|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.</ref>

However, the term "domain" was occasionally used to identify closely related but slightly different concepts. For example, in his influential [[monograph]]s on [[elliptic partial differential equation]]s, [[Carlo Miranda]] uses the term "region" to identify an open connected set,<ref name="M&Mp1">See {{harvs|last= Miranda|year1=1955|year2=1970|loc1=p. 1|loc2=p. 2}}.</ref><ref>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 [[Field (agriculture)|its meaning in agriculture]]: in the second edition of the book, Zane C. Motteler appropriately translates this term as "region".</ref> and reserves the term "domain" to identify an internally connected,<ref>An internally connected set is a set whose interior is connected.</ref> [[perfect set]], each point of which is an accumulation point of interior points,<ref name="M&Mp1"/> following his former master [[Mauro Picone]]:<ref>See {{harv|Picone|1923|p=66}}.</ref> according to this convention, if a set {{math|''A''}} is a region then its [[Closure (topology)|closure]] {{math|''{{overline|A}}''}} is a domain.<ref name="M&Mp1" />

==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==
{{notelist|29em}}
{{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 }}<br>{{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 }} <!-- link goes to 1964 third edition -->
* {{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=http://mathematica.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}} <br> {{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]]
[[Category:Partial differential equations]]
[[Category:Topology]]