# Domain (mathematical analysis)

## Connected open subset of a topological space / From Wikipedia, the free encyclopedia

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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]}