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Open Set

Basic definition

An open set is a set for which every point in the set has a neighborhood lying in the set. An open set is the complement of a closed set and. An open interval is an example of an open set.

Detailed 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 left bracketing bar x - x_0 right bracketing bar <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.

Related terms

ball | Borel set | closed set | empty set | neighborhood | open ball | open disk | open interval | open neighborhood

Educational grade level

college level

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