Domain (mathematical analysis)

In mathematical analysis, a domain or region is a non-empty connected open set in a topological space, in particular any non-empty connected open subset of the real coordinate space Rⁿ or the complex coordinate space Cⁿ. 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, some use the term region, some use both terms interchangeably, and some define the two terms slightly differently; some avoid ambiguity by sticking with a phrase such as non-empty connected open subset.