Open Set

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open set

[′ō·pən ‚set]
A set included in a topology; equivalently, a set which is a neighborhood of each of its points; a topology on a space is determined by a collection of subsets which are called open.

Open Set


a point set that does not contain the limit points of its complement. Every point of an open set is an interior point, that is, it has a neighborhood entirely contained in the open set. Together with closed sets, open sets play an important role in the theory of functions, in topology, and in other branches of mathematics. Any nonempty open set on a line is an open interval or a sum of an at most countable number of open intervals.

The concept of open set can be applied in an n-dimensional Euclidean space and also in an arbitrary metric or topological space. The intersection of a finite number of open sets is an open set, as is the union of any number of open sets. Connected open sets are called domains. Any topological space can be defined by specifying its open sets. If a topological space is given by a system of its closed sets, then the open sets are defined in it as the complements of the closed sets.