Open set

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Example: The blue circle represents the set of points (x, y) satisfying x + y = r. The red disk represents the set of points (x, y) satisfying x + y < r. The red set is an open set, the blue set is its boundary set, and the union of the red and blue sets is a closed set. Red blue circle.svg
Example: The blue circle represents the set of points (x, y) satisfying x + y = r. The red disk represents the set of points (x, y) satisfying x + y<r. The red set is an open set, the blue set is its boundary set, and the union of the red and blue sets is a closed set.

In mathematics, open sets are a generalization of open intervals in the real line. In a metric space—that is, when a distance is defined—open sets are the sets that, with every point P, contain all points that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P).

Contents

More generally, one defines open sets as the members of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, every subset can be open (the discrete topology), or no set can be open but the space itself and the empty set (the indiscrete topology).

In practice, however, open sets are usually chosen to provide a notion of nearness that is similar to that of metric spaces, without having a measure of distance defined. In particular, a topology allows defining properties such as continuity, connectedness, and compactness, which were originally defined by means of a distance.

The most common case of a topology without any distance is given by manifolds, which are topological spaces that, near each point, resemble an open set of a Euclidean space, but on which no distance is defined in general. Less intuitive topologies are used in other branches of mathematics; for example, the Zariski topology, which is fundamental in algebraic geometry and scheme theory.

Motivation

Intuitively, an open set provides a method to distinguish two points. For example, if about one of two points in a topological space, there exists an open set not containing the other (distinct) point, the two points are referred to as topologically distinguishable. In this manner, one may speak of whether two points, or more generally two subsets, of a topological space are "near" without concretely defining a distance. Therefore, topological spaces may be seen as a generalization of spaces equipped with a notion of distance, which are called 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) = |xy|. Therefore, given a real number x, one can speak of the set of all points close to that real number; that is, within ε of 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 than 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. It may help in this case to think of the measure as being a binary condition: all things in R are equally close to 0, while any item that is not in R is not close to 0.

In general, one refers to the family of sets containing 0, used to approximate 0, as 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

Several definitions are given here, in an increasing order of technicality. Each one is a special case of the next one.

Euclidean space

A subset of the Euclidean n-space Rn is open if, for every point x in , there exists a positive real number ε (depending on x) such that a point in Rn belongs to as soon as its Euclidean distance from x is smaller than ε. [1] Equivalently, a subset of Rn is open if every point in is the center of an open ball contained in

Metric space

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 satisfying d(x, y) < ε, y also belongs to U. Equivalently, U is open if every point in U has a neighborhood contained in U.

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

Topological space

A topological space is a set on which a topology is defined, which consists of a collection of subsets that are said to be open, and satisfy the axioms given below.

More precisely, let be a set. A family of subsets of is a topology on , and the elements of are the open sets of the topology if

Infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form where is a positive integer, is the set which is not open in the real line.

A metric space is a topological space, whose topology consists of the collection of all subsets that are unions of open balls. There are, however, topological spaces that are not metric spaces.

Special types of open sets

Clopen sets and non-open and/or non-closed sets

A set might be open, closed, both, or neither. In particular, open and closed sets are not mutually exclusive, meaning that it is in general possible for a subset of a topological space to simultaneously be both an open subset and a closed subset. Such subsets are known as clopen sets . Explicitly, a subset of a topological space is called clopen if both and its complement are open subsets of ; or equivalently, if and

In any topological space the empty set and the set itself are always open. These two sets are the most well-known examples of clopen subsets and they show that clopen subsets exist in every topological space. To see why is clopen, begin by recalling that the sets and are, by definition, always open subsets (of ). Also by definition, a subset is called closed if (and only if) its complement in which is the set is an open subset. Because the complement (in ) of the entire set is the empty set (i.e. ), which is an open subset, this means that is a closed subset of (by definition of "closed subset"). Hence, no matter what topology is placed on the entire space is simultaneously both an open subset and also a closed subset of ; said differently, is always a clopen subset of Because the empty set's complement is which is an open subset, the same reasoning can be used to conclude that is also a clopen subset of

Consider the real line endowed with its usual Euclidean topology, whose open sets are defined as follows: every interval of real numbers belongs to the topology, every union of such intervals, e.g. belongs to the topology, and as always, both and belong to the topology.

If a topological space is endowed with the discrete topology (so that by definition, every subset of is open) then every subset of is a clopen subset. For a more advanced example reminiscent of the discrete topology, suppose that is an ultrafilter on a non-empty set Then the union is a topology on with the property that every non-empty proper subset of is either an open subset or else a closed subset, but never both; that is, if (where ) then exactly one of the following two statements is true: either (1) or else, (2) Said differently, every subset is open or closed but the only subsets that are both (i.e. that are clopen) are and

Regular open sets

A subset of a topological space is called a regular open set if or equivalently, if where (resp. ) denotes the topological boundary (resp. interior, closure) of in A topological space for which there exists a base consisting of regular open sets is called a semiregular space . A subset of is a regular open set if and only if its complement in is a regular closed set, where by definition a subset of is called a regular closed set if or equivalently, if Every regular open set (resp. regular closed set) is an open subset (resp. is a closed subset) although in general, [note 1] the converses are not true.

Properties

The union of any number of open sets, or infinitely many open sets, is open. [2] The intersection of a finite number of open sets is open. [2]

A complement of an open set (relative to the space that the topology is defined on) is called a closed set. A set may be both open and closed (a clopen set). The empty set and the full space are examples of sets that are both open and closed. [3]

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 maximum (ordered under inclusion) 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.

A function between two topological spaces and is continuous if the preimage of every open set in is open in The function is called open if the image of every open set in is open in

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 defined relative to a particular topology

Whether a set is open depends on the topology under consideration. Having opted for greater brevity over greater clarity, we refer to a set X endowed with a topology as "the topological space X" rather than "the topological space ", despite the fact that all the topological data is contained in If there are two topologies on the same set, a set U that is open in the first topology might fail to be open in the second topology. For example, if X is any topological space and Y is any subset of X, the set Y can be given its own topology (called the 'subspace topology') defined by "a set U is open in the subspace topology on Y if and only if U is the intersection of Y with an open set from the original topology on X." This potentially introduces new open sets: if V is open in the original topology on X, but isn't open in the original topology on X, then is open in the subspace topology on Y.

As a concrete example of this, if U is defined as the set of rational numbers in the interval 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 (because U contains no non-rational numbers).

Generalizations of open sets

Throughout, will be a topological space.

A subset of a topological space is called:

Using the fact that

    and    

whenever two subsets satisfy the following may be deduced:

Moreover, a subset is a regular open set if and only if it is preopen and semi-closed. [5] The intersection of an α-open set and a semi-preopen (resp. semi-open, preopen, b-open) set is a semi-preopen (resp. semi-open, preopen, b-open) set. [5] Preopen sets need not be semi-open and semi-open sets need not be preopen. [5]

Arbitrary unions of preopen (resp. α-open, b-open, semi-preopen) sets are once again preopen (resp. α-open, b-open, semi-preopen). [9] However, finite intersections of preopen sets need not be preopen. [8] The set of all α-open subsets of a space forms a topology on that is finer than [4]

A topological space is Hausdorff if and only if every compact subspace of is θ-closed. [8] A space is totally disconnected if and only if every regular closed subset is preopen or equivalently, if every semi-open subset is preopen. Moreover, the space is totally disconnected if and only if the closure of every preopen subset is open. [4]

See also

Notes

  1. One exception if the if is endowed with the discrete topology, in which case every subset of is both a regular open subset and a regular closed subset of

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