It has been suggested that Region (mathematical analysis) be merged into this article. (Discuss) Proposed since September 2021. |

In mathematical analysis, a **domain** or **region** is a non-empty connected open set in a topological space X, in particular any non-empty connected open subset of the real coordinate space **R**^{n} or the complex coordinate space **C**^{n}. 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, *C*^{1} 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 **C**^{n}.

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] }

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,^{ [4] }^{ [5] } and reserves the term "domain" to identify an internally connected,^{ [6] } perfect set, each point of which is an accumulation point of interior points,^{ [4] } following his former master Mauro Picone:^{ [7] } according to this convention, if a set *A* is a region then its closure *A* is a domain.^{ [4] }

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.)
^{ [8] }

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*.^{ [9] }

- ↑ 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.*" - 1 2 3 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) .
- ↑ Erwin Kreyszig (1993)
*Advanced Engineering Mathematics*, 7th edition, p. 720, John Wiley & Sons, ISBN 0-471-55380-8 - ↑ 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

In mathematics, a **holomorphic function** is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space **C**^{n}. The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (*analytic*). Holomorphic functions are the central objects of study in complex analysis.

In complex analysis, the **Riemann mapping theorem** states that if *U* is a non-empty simply connected open subset of the complex number plane **C** which is not all of **C**, then there exists a biholomorphic mapping *f* from *U* onto the open unit disk

In mathematics, the **domain** or **set of departure** of a function is the set into which all of the input of the function is constrained to fall. It is the set X in the notation *f*: *X* → *Y*, and is alternatively denoted as . Since a (total) function is defined on its entire domain, its domain coincides with its domain of definition. However, this coincidence is no longer true for a partial function, since the domain of definition of a partial function can be a proper subset of the domain.

In mathematics, a **conformal map** is a function that locally preserves angles, but not necessarily lengths.

In topology, a **Jordan curve**, sometimes called a *plane simple closed curve*, is a non-self-intersecting continuous loop in the plane. The **Jordan curve theorem** asserts that every Jordan curve divides the plane into an "interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior points, so that every continuous path connecting a point of one region to a point of the other intersects with that loop somewhere. While the statement of this theorem seems to be intuitively obvious, it takes some ingenuity to prove it by elementary means. *"Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it."*. More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces.

In the mathematical field of measure theory, an **outer measure** or **exterior measure** is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures. Carathéodory's work on outer measures found many applications in measure-theoretic set theory, and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff dimension. Outer measures are commonly used in the field of geometric measure theory.

In mathematics, the **total variation** identifies several slightly different concepts, related to the structure of the codomain of a function or a measure. For a real-valued continuous function *f*, defined on an interval [*a*, *b*] ⊂ **R**, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation *x* ↦ *f*(*x*), for *x* ∈ [*a*, *b*]. Functions whose total variation is finite are called **functions of bounded variation**.

In mathematics, **Carathéodory's theorem** may refer to one of a number of results of Constantin Carathéodory:

In the theory of several complex variables and complex manifolds in mathematics, a **Stein manifold** is a complex submanifold of the vector space of *n* complex dimensions. They were introduced by and named after Karl Stein (1951). A **Stein space** is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry.

In mathematics, a **Kleinian group** is a discrete subgroup of PSL(2, **C**). The group PSL(2, **C**) of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic space **H**^{3}, and as orientation-preserving conformal maps of the open unit ball *B*^{3} in **R**^{3} to itself. Therefore, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.

In mathematics, with special application to complex analysis, a *normal family* is a pre-compact subset of the space of continuous functions. Informally, this means that the functions in the family are not widely spread out, but rather stick together in a somewhat "clustered" manner. Sometimes, if each function in a normal family *F* satisfies a particular property , then the property also holds for each limit point of the set *F*.

In mathematics, **Carathéodory's theorem** is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, first proved in 1913, states that the conformal mapping sending the unit disk to the region in the complex plane bounded by a Jordan curve extends continuously to a homeomorphism from the unit circle onto the Jordan curve. The result is one of Carathéodory's results on prime ends and the boundary behaviour of univalent holomorphic functions.

In mathematics, a **mixed boundary condition** for a partial differential equation defines a boundary value problem in which the solution of the given equation is required to satisfy different boundary conditions on disjoint parts of the boundary of the domain where the condition is stated. Precisely, in a mixed boundary value problem, the solution is required to satisfy a Dirichlet or a Neumann boundary condition in a mutually exclusive way on disjoint parts of the boundary.

In mathematics, a **locally integrable function** is a function which is integrable on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to *L*^{p} spaces, but its members are not required to satisfy any growth restriction on their behavior at the boundary of their domain : in other words, locally integrable functions can grow arbitrarily fast at the domain boundary, but are still manageable in a way similar to ordinary integrable functions.

**Renato Caccioppoli** was an Italian mathematician, known for his contributions to mathematical analysis, including the theory of functions of several complex variables, functional analysis, measure theory.

In mathematics, especially potential theory, **harmonic measure** is a concept related to the theory of harmonic functions that arises from the solution of the classical Dirichlet problem.

In mathematics, the **prime end** compactification is a method to compactify a topological disc by adding the boundary circle in an appropriate way.

In topology and related areas of mathematics, a subset *A* of a topological space *X* is called **dense** if every point *x* in *X* either belongs to *A* or is a limit point of *A*; that is, the closure of *A* constitutes the whole set *X*. Informally, for every point in *X*, the point is either in *A* or arbitrarily "close" to a member of *A* — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it.

In mathematical analysis, the word * region* usually refers to a subset of or that is open, simply connected and non-empty. A

In mathematics, a **planar Riemann surface** is a Riemann surface sharing the topological properties of a connected open subset of the Riemann sphere. They are characterized by the topological property that the complement of every closed Jordan curve in the Riemann surface has two connected components. An equivalent characterization is the differential geometric property that every closed differential 1-form of compact support is exact. Every simply connected Riemann surface is planar. The class of planar Riemann surfaces was studied by Koebe who proved in 1910 as a generalization of the uniformization theorem that every such surface is conformally equivalent to either the Riemann sphere or the complex plane with slits parallel to the real axis removed.

- 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) [1955],
*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*"). - Solomentsev, E.D. (2001) [1994], "Domain",
*Encyclopedia of Mathematics*, EMS Press

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