The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking (see below for a more intuitive discussion), a set U is open if any point x in U can be moved in any "direction" and still be in the set 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 manifold in differential topology where each point within the space is contained in an open set that is homeomorphic to an open ball in a finite-dimensional Euclidean space.
Point-set topology is the area of mathematics concerned with general topological spaces, and the relations between them. In the category of topological spaces, morphisms are continuous functions between topological spaces. Continuous functions are readily observed to preserve topological structure, as they map "points close together" to "points close together"; that is, they preserve the structure of open sets defined on the space.
In metric topology, one can concretely define a distance function between two points, and thus metric spaces also have a topology, i.e. a certain structure of open sets defined on them. Thus as opposed to the pure topological invariants, metric topology deals with isometries and the like; that is, distance preserving maps. In this case, the idea of an open set is used as an organizational tool rather than an object of study. From the topological point of view, metric spaces are fairly well understood, although many open problems still remain in metrizability theory.
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