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In mathematical analysis, the word region usually refers to a subset of or 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.)
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.
- 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.