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 .

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