Open Set

Let  be a subset of a metric space. Then the set  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 .

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.