In topology, a set U is called an open set if it does not contain any of its boundary points. When dealing with metric spaces, there is a well-defined distance between any two points. A subset U of a metric space is open if, for every point p in U, there is some (possibly very small) positive distance such that every point which is at least this close to p is also contained in U.
The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined. Concepts that use notions of nearness, such as the continuity of functions, can be translated into the language of open sets.
In point-set topology, open sets are used to distinguish between points and subsets of a space. The degree to which any two points can be separated is specified by the separation axioms. The collection of all open sets in a space defines the topology of the space. Functions from one topological space to another that preserve the topology are the continuous functions. Although open sets and the topologies that they comprise are of central importance in point-set topology, they are also used as an organizational tool in other important branches of mathematics. Examples of topologies include the Zariski topology in algebraic geometry that reflects the algebraic nature of varieties, and the topology on a differential mani
Famous quotes containing the words open and/or set:
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—John Gay (1685–1732)
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—Vladimir Nabokov (1899–1977)