Domain (mathematical analysis)
In mathematical analysis, a domain is any connected open subset of a finite-dimensional vector space. 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.
In complex analysis, a complex domain (or simply domain) is any connected open subset of the complex plane . 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 .
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. 
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). Hahn also remarks that the word "Gebiet" ("Domain") was occasionally previously used as a synonym of open set. 
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. 
- 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.
- 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."
- 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 1922, p. 66) harv error: no target: CITEREFPicone1922 ( help).
- Carathéodory, Constantin (1918), Vorlesungen über reelle Funktionen (in German) (1st ed.), Leipzig und Berlin: B. G. Teubner Verlag, pp. X+704, JFM 46.0376.12, MR 0225940 (the MR review refers to the third corrected edition).
- Hahn, Hans (1921), Theorie der reellen Funktionen. Erster Band (in German), Vienna: Springer-Verlag, pp. VII+600, doi: 10.1007/978-3-642-52624-4, hdl: 2027/pst.000003378601, ISBN 978-3-642-52570-4, JFM 48.0261.09 (freely available at the Internet Archive).
- Steven G. Krantz & Harold R. Parks (1999) The Geometry of Domains in Space, Birkhäuser ISBN 0-8176-4097-5.
- Miranda, Carlo (1955), Equazioni alle derivate parziali di tipo ellittico, Ergebnisse der Mathematik und ihrer Grenzgebiete – Neue Folge (in Italian), Heft 2 (1st ed.), Berlin – Göttingen – New York: Springer Verlag, pp. VIII+222, MR 0087853, Zbl 0065.08503.
- Miranda, Carlo (1970) , Partial Differential Equations of Elliptic Type, Ergebnisse der Mathematik und ihrer Grenzgebiete – 2 Folge, Band 2 (2nd Revised ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. XII+370, ISBN 978-3-540-04804-6, MR 0284700, Zbl 0198.14101, translated from the Italian by Zane C. Motteler.
- Picone, Mauro (1923), Lezioni di analisi infinitesimale (PDF), Volume 1 (in Italian), Parte Prima – La Derivazione, Catania: Circolo matematico di Catania, pp. xii+351, JFM 49.0172.07 (Review of the whole volume I) (available from the " Edizione Nazionale Mathematica Italiana").