Open set

Example: The points (x, y) satisfying x2 + y2 = r2 are colored blue. The points (x, y) satisfying x2 + y2 < r2 are colored red. The red points form an open set. The union of the red and blue points is a closed set.

The concept of an open set is fundamental to many areas of mathematics, especially including 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.

Contents

Motivation

Intuitively, an open set provides a method to distinguish two points. For example, if about one point in a topological space there exists an open set not containing another (distinct) point, the two points are referred to as topologically distinguishable. In this manner, one may speak of whether two subsets of a topological space are "near" without concretely defining a metric on the topological space. Therefore, topological spaces may be seen as a generalization of metric spaces.

In the set of all real numbers, one has the natural Euclidean metric; that is, a function which measures the distance between two real numbers: d(x, y) = |x - y|. Therefore, given a real number, one can speak of the set of all points close to that real number; that is, within ε of that real number (refer to this real number as x). In essence, points within ε of x approximate x to an accuracy of degree ε. Note that ε > 0 always but as ε becomes smaller and smaller, one obtains points that approximate x to a higher and higher degree of accuracy. For example, if x = 0 and ε = 1, the points within ε of x are precisely the points of the interval (-1, 1); that is, the set of all real numbers between -1 and 1. However, with ε = 0.5, the points within ε of x are precisely the points of (-0.5, 0.5). Clearly, these points approximate x to a greater degree of accuracy compared to when ε = 1.

The previous discussion shows, for the case x = 0, that one may approximate x to higher and higher degrees of accuracy by defining ε to be smaller and smaller. In particular sets of the form (-ε, ε) give us a lot of information about points close to x = 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x. This innovative idea has far-reaching consequences; in particular, by defining different collections of sets containing 0 (distinct from the sets (-ε, ε)), one may find different results regarding the distance between 0 and other real numbers. For example, if we were to define R as the only such set for "measuring distance", all points are close to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a member of R. Thus, we find that in some sense, every real number is distance 0 away from 0!

In general, one refers to the family of sets containing 0, used to approximate 0, a neighborhood basis; a member of this neighborhood basis is referred to as an open set. In fact, one may generalize these notions to an arbitrary set (X); rather than just the real numbers. In this case, given a point (x) of that set, one may define a collection of sets "around" (that is, containing) x, used to approximate x. Of course, this collection would have to satisfy certain properties (known as axioms) for otherwise we may not have a well-defined method to measure distance. For example, every point in X should approximate x to some degree of accuracy. Thus X should be in this family. Once we begin to define "smaller" sets containing x, we tend to approximate x to a greater degree of accuracy. Bearing this in mind, one may define the remaining axioms that the family of sets about x is required to satisfy.

Definitions

The concept of open sets can be formalized with various degrees of generality, for example:

Geometric

A point set in Rn is called open when every point P of the set is an interior point.

Euclidean space

A subset U of the Euclidean n-space Rn is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in Rn whose Euclidean distance from x is smaller than ε, y also belongs to U. Equivalently, a subset U of Rn is open if every point in U has a neighbourhood in Rn contained in U.

Metric spaces

A subset U of a metric space (M, d) is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with d(x, y) < ε, y also belongs to U. Equivalently, U is open if every point in U has a neighbourhood contained in U.

This generalises the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.

Topological spaces

If a nonempty set X has a collection of subsets T that is a topological space, then any member of T is an open set.

Note that infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form (−1/n, 1/n), where n is a positive integer, is the set {0} which is closed in the real line. Sets that can be constructed as the intersection of countably many open sets are denoted Gδ sets.

The topological definition of open sets generalises the metric space definition: If one begins with a metric space and defines open sets as before, then the family of all open sets is a topology on the metric space. Every metric space is therefore, in a natural way, a topological space. There are, however, topological spaces that are not metric spaces.

Properties

Uses

Open sets have a fundamental importance in topology. The concept is required to define and make sense of topological space and other topological structures that deal with the notions of closeness and convergence for spaces such as metric spaces and uniform spaces.

Every subset A of a topological space X contains a (possibly empty) open set; the largest such open set is called the interior of A. It can be constructed by taking the union of all the open sets contained in A.

Given topological spaces X and Y, a function f from X to Y is continuous if the preimage of every open set in Y is open in X. The function f is called open if the image of every open set in X is open in Y.

An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals.

Notes and cautions

"Open" is not an absolute definition

Note that whether a given set U is open depends on the surrounding space. More precisely, if U is a subset of X and X is a subspace of Y, then U may be open in X but not Y. For instance, if U is defined as the set of rational numbers in the interval (0, 1), then U is an open subset of the rational numbers, but not of the real numbers. This is because when the surrounding space is the rational numbers, for every point x in U, there exists a positive number a such that all rational points within distance a of x are also in U. On the other hand, when the surrounding space is the reals, then for every point x in U there is no positive a such that all real points within distance a of x are in U (since U contains no non-rational numbers).

"Open" is not the opposite of "closed"

Rather, a set is open if and only if its complement is closed. Indeed, some spaces (the disconnected ones) have nontrivial subsets which are both open and closed (called clopen sets). For instance, the set of all rational numbers smaller than √2 is clopen in the rationals. Moreover, many spaces have subsets which are neither open nor closed, such as the half-open interval (0, 1] in the real numbers.

See also

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