A set in a metric space is open if every point in has a neighborhood lying in the set. An open set of radius and center is the set of all points such that , and is denoted . In one-space, the open set is an open interval. In two-space, the open set is a disk. In three-space, the open set is a ball.

More generally, given a topology (consisting of a set and a collection of subsets ), a set is said to be open if it is in . Therefore, while it is not possible for a set to be both finite and open in the topology of the real line (a single point is a closed set), it is possible for a more general topological set to be both finite and open.

The complement of an open set is a closed set. It is possible for a set to be neither open nor closed, e.g., the half-closed interval .

See also

Ball, Borel Set, Closed Set, Empty Set, Neighborhood, Open Ball, Open Disk, Open Interval, Open Neighborhood Explore this topic in the MathWorld classroom

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References

Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991.Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 3, 1999.

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Open Set

Cite this as:

Weisstein, Eric W. "Open Set." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OpenSet.html

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