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In mathematics, an open set is a generalization of an open interval in the real line.
In a metric space (a set along with a distance defined between any two points), an open set is a set that, along with every point P, contains all points that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P).
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 (the discrete topology), or no subset can be open except the space itself and the empty set (the indiscrete topology).^{ [1]}
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.
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) = |x − y|. 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.
Several definitions are given here, in an increasing order of technicality. Each one is a special case of the next one.
A subset of the Euclidean n-space R^{n} is open if, for every point x in , there exists a positive real number ε (depending on x) such that any point in R^{n} whose Euclidean distance from x is smaller than ε belongs to .^{ [2]} Equivalently, a subset of R^{n} is open if every point in is the center of an open ball contained in
An example of a subset of R that is not open is the closed interval [0,1], since neither 0 - ε nor 1 + ε belongs to [0,1] for any ε > 0, no matter how small.
A subset U of a metric space (M, d) is called open if, for any point x in U, there exists a real number ε > 0 such that any point satisfying d(x, y) < ε 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.
A topology on a set X is a set of subsets of X with the properties below. Each member of is called an open set.^{ [3]}
X 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.
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 clopen. 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
A subset of a topological space is called a regular open set if or equivalently, if , where , , and denote, respectively, the topological boundary, interior, and 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.
The union of any number of open sets, or infinitely many open sets, is open.^{ [4]} The intersection of a finite number of open sets is open.^{ [4]}
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.^{ [5]}
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.^{ [6]}
A function between two topological spaces and is continuous if the preimage of every open set in is open in ^{ [7]} 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.
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."^{ [8]} 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).
Throughout, will be a topological space.
A subset of a topological space is called:
The complement of a preopen set is called preclosed.
The complement of a β-open set is called β-closed.
The complement of a sequentially open set is called sequentially closed. A subset is sequentially closed in if and only if is equal to its sequential closure, which by definition is the set consisting of all for which there exists a sequence in that converges to (in ).
Using the fact that
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.^{ [10]} 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.^{ [10]} Preopen sets need not be semi-open and semi-open sets need not be preopen.^{ [10]}
Arbitrary unions of preopen (resp. α-open, b-open, semi-preopen) sets are once again preopen (resp. α-open, b-open, semi-preopen).^{ [10]} However, finite intersections of preopen sets need not be preopen.^{ [13]} The set of all α-open subsets of a space forms a topology on that is finer than ^{ [9]}
A topological space is Hausdorff if and only if every compact subspace of is θ-closed.^{ [13]} 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.^{ [9]}