Domain (mathematical analysis)

In mathematical analysis, a domain or region is a non-empty, connected, and open set in a topological space, in particular any non-empty connected open subset of the real coordinate space $R n$ or the complex coordinate space $C n$ . A connected open subset of coordinate space is frequently used for the domain of a function, but in general, functions 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,^[1] some use the term region,^[2] some use both terms interchangeably,^[3] and some define the two terms slightly differently;^[4] some avoid ambiguity by sticking with a phrase such as non-empty connected open subset.^[5]

^ For instance (Sveshnikov & Tikhonov 1978, §1.3 pp. 21–22).
^ For instance (Churchill 1948, §1.9 pp. 16–17); (Ahlfors 1953, §2.2 p. 58); (Rudin 1974, §10.1 p. 213) reserves the term domain for the domain of a function; (Carathéodory 1964, p. 97) uses the term region for a connected open set and the term continuum for a connected closed set.
^ For instance (Townsend 1915, §10, p. 20); (Carrier, Krook & Pearson 1966, §2.2 p. 32).
^ For instance (Churchill 1960, §1.9 p. 17), who does not require that a region be connected or open.
^ For instance (Dieudonné 1960, §3.19 pp. 64–67) generally uses the phrase open connected set, but later defines simply connected domain (§9.7 p. 215); Tao, Terence (2016). "246A, Notes 2: complex integration"., also, (Bremermann 1956) called the region an open set and the domain a concatenated open set.

[1] For instance (Sveshnikov & Tikhonov 1978, §1.3 pp. 21–22).

[2] For instance (Churchill 1948, §1.9 pp. 16–17); (Ahlfors 1953, §2.2 p. 58); (Rudin 1974, §10.1 p. 213) reserves the term domain for the domain of a function; (Carathéodory 1964, p. 97) uses the term region for a connected open set and the term continuum for a connected closed set.

[3] For instance (Townsend 1915, §10, p. 20); (Carrier, Krook & Pearson 1966, §2.2 p. 32).

[4] For instance (Churchill 1960, §1.9 p. 17), who does not require that a region be connected or open.

[5] For instance (Dieudonné 1960, §3.19 pp. 64–67) generally uses the phrase open connected set, but later defines simply connected domain (§9.7 p. 215); Tao, Terence (2016). "246A, Notes 2: complex integration"., also, (Bremermann 1956) called the region an open set and the domain a concatenated open set.

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