Open set

In general topology and mathematical analysis, an open set is a generalization of an open interval in the real line.
In a metric space, an open set is a set that, with every point in it, contains all points of the metric space that are sufficiently near to .
More generally, an open set is a member 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, or no subset can be open except the space itself and the empty set.
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 notion 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 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:. 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 ; 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. 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 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, one may find different results regarding the distance between 0 and other real numbers. For example, if we were to define as the only such set for "measuring distance", all points would be close to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a member of. 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 are equally close to 0, while any item that is not in 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 ; rather than just the real numbers. In this case, given a point of that set, one may define a collection of sets "around" x, used to approximate x. Of course, this collection would have to satisfy certain properties 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 -space is open if, for every point in, there exists a positive real number such that any point in whose Euclidean distance from is smaller than belongs to. Equivalently, a subset of is open if every point in is the center of an open ball contained in
An example of a subset of that is not open is the closed interval, since neither nor belongs to for any, no matter how small.

Metric space

A subset U of a metric space is called open if, for any point x in U, there exists a real number ε > 0 such that any point satisfying 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 topology on a set is a set of subsets of with the following properties:

and.
Any union of sets in belong to : if then
Any finite intersection of sets in belong to : for a positive integer, if then

Each member of is called an open set. The set together with is called a topological space.
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.

Properties

The union of any number of open sets, or infinitely many open sets, is open. The intersection of a finite number of open sets is open.
A complement of an open set is called a closed set. A set may be both open and closed. The empty set and the full space are examples of sets that are both open and closed.
A set can never been considered as open by itself. This notion is relative to a containing set and a specific topology on it.
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 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 ε such that all points within distance ε 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 positive ε such that all points within distance ε of x are in U.

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 open set; the maximum 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 if the preimage of every open set in is open in
The function is called 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.

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 a closed subset. Such subsets are known as . Explicitly, a subset of a topological space is called if both and its complement are open subsets of ; or equivalently, if and
In topological space the empty set and the set itself are always clopen. These two sets are the most well-known examples of clopen subsets and they show that clopen subsets exist in topological space. To see, it suffices to remark that, by definition of a topology, and are both open, and that they are also closed, since each is the complement of the other.
The open sets of the usual Euclidean topology of the real line are the empty set, the open intervals and every union of open intervals.

The interval is open in by definition of the Euclidean topology. It is not closed since its complement in is which is not open; indeed, an open interval contained in cannot contain, and it follows that cannot be a union of open intervals. Hence, is an example of a set that is open but not closed.
By a similar argument, the interval is a closed subset but not an open subset.
Finally, neither nor its complement are open ; this means that is neither open nor closed.

If a topological space is endowed with the discrete topology 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 non-empty proper subset of is an open subset or else a closed subset, but never both; that is, if then of the following two statements is true: either or else, Said differently, subset is open or closed but the subsets that are both are and