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Open Set

  • Definition: Let S be a subset of a metric space. Then the set S is open if every point in S has a neighborhood lying in the set. An open set of radius r and center x_0 is the set of all points x such that |x-x_0|< r, and is denoted D_r(x_0). 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 X and a collection of subsets T), a set is said to be open if it is in T. 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 (0,1].


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