Region (mathematics)
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Short description: Mathematical subset of a space
In mathematical analysis, the word region usually refers to a subset of [math]\displaystyle{ \R^n }[/math] or [math]\displaystyle{ \Complex^n }[/math] that is open (in the standard Euclidean topology), simply connected and non-empty. A closed region is sometimes defined to be the closure of a region.
Regions and closed regions are often used as domains of functions or differential equations.
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.)[1]
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.[2]
See also
- Area
- Curve
- Face (geometry)
- Interval (mathematics)
- Jordan curve theorem
- Line segment
- Locus (mathematics)
- Neighbourhood (mathematics)
- Point (geometry)
- Riemann mapping theorem
- Shape
- Simple polygon
Notes
- ↑ 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
References
- Ruel V. Churchill (1960) Complex variables and applications, 2nd edition, §1.9 Regions in the complex plane, pp. 16 to 18, McGraw-Hill
- Constantin Carathéodory (1954) Theory of Functions of a Complex Variable, v. I, p. 97, Chelsea Publishing.
- Howard Eves (1966) Functions of a Complex Variable, p. 105, Prindle, Weber & Schmidt.
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