# Open set

In mathematics, particularly in topology, an

**open set**is an abstract concept generalizing the idea of an open interval in the real line. The simplest example is in metric spaces, where open sets can be defined as those sets which contain a ball around each of their points ; however, an open set, in general, can be very abstract: any collection of sets can be called open, as long as the union of an arbitrary number of open sets in the collection is open, the intersection of a finite number of open sets is open, and the space itself is open. These conditions are very loose, and they allow enormous flexibility in the choice of open sets. In the two extremes, every set can be open, or no set can be open but the space itself and the empty set.

In practice, however, open sets are usually chosen to be similar to the open intervals of the real line. 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. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, which use notions of nearness, can be defined using these open sets.

Each choice of open sets for a space is called a topology. 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.

## 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 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:. 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

**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 ; 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 a point in belongs to as soon as its Euclidean distance from is smaller than. Equivalently, a subset of is open if every point in is the center of an open ball contained in.

### Metric space

A subset*U*of a metric space 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

*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

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.

## 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 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.

### "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

*T*as "the topological space

*X*" rather than "the topological space ", despite the fact that all the topological data is contained in

*T*. 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

*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*.

### Open and closed are not mutually exclusive

A set might be open, closed, both, or neither.For example, we'll use the real line with its usual topology, which is defined as follows: every interval of real numbers belongs to the topology, and every union of such intervals, e.g., belongs to the topology.

- In
*any*topology, the entire set*X*is declared open by definition, as is the empty set. Moreover, the complement of the entire set*X*is the empty set; since*X*has an open complement, this means by definition that*X*is closed. Hence, in any topology, the entire space is simultaneously open and closed. - The interval is open because it belongs to the Euclidean topology. If were to have an open complement, it would mean by definition that were closed. But does not have an open complement; its complement is, which does
*not*belong to the Euclidean topology since it is not a union of intervals of the form. Hence, is an example of a set that is open but not closed. - By a similar argument, the interval is closed but not open.
- Finally, since neither nor its complement belongs to the Euclidean topology, this means that is neither open nor closed.