In mathematics, an open set is a generalization of an open interval in the real line.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Red_blue_circle.svg/220px-Red_blue_circle.svg.png)
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.
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) = |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.
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 any point in Rn whose Euclidean distance from x is smaller than ε belongs to
.[2] Equivalently, a subset
of Rn 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.
Metric space
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.
Topological 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]
and
- Any union of sets in
belong to
: if
then
- Any finite intersection of sets in
belong to
: if
then
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.
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 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.
- The interval
is open in
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 open intervals of the form
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, since neither
nor its complement
belongs to the Euclidean topology (because it can not be written as a union of intervals of the form
), this means that
is neither open nor closed.
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
,
, 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.
Properties
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]
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.[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.
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."[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).
Generalizations of open sets
Throughout, will be a topological space.
A subset of a topological space
is called:
- α-open if
, and the complement of such a set is called α-closed.[9]
- preopen, nearly open, or locally dense if it satisfies any of the following equivalent conditions:
[10]
- There exists subsets
such that
is open in
is a dense subset of
and
[10]
- There exists an open (in
) subset
such that
is a dense subset of
[10]
The complement of a preopen set is called preclosed.
- b-open if
. The complement of a b-open set is called b-closed.[9]
- β-open or semi-preopen if it satisfies any of the following equivalent conditions:
The complement of a β-open set is called β-closed.
- sequentially open if it satisfies any of the following equivalent conditions:
- Whenever a sequence in
converges to some point of
then that sequence is eventually in
Explicitly, this means that if
is a sequence in
and if there exists some
is such that
in
then
is eventually in
(that is, there exists some integer
such that if
then
).
is equal to its sequential interior in
which by definition is the set
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
).
- Whenever a sequence in
- almost open and is said to have the Baire property if there exists an open subset
such that
is a meager subset, where
denotes the symmetric difference.[11]
- The subset
is said to have the Baire property in the restricted sense if for every subset
of
the intersection
has the Baire property relative to
.[12]
- The subset
- semi-open if
. The complement in
of a semi-open set is called a semi-closed set.[13]
- The semi-closure (in
) of a subset
denoted by
is the intersection of all semi-closed subsets of
that contain
as a subset.[13]
- The semi-closure (in
- semi-θ-open if for each
there exists some semiopen subset
of
such that
[13]
- θ-open (resp. δ-open) if its complement in
is a θ-closed (resp. δ-closed) set, where by definition, a subset of
is called θ-closed (resp. δ-closed) if it is equal to the set of all of its θ-cluster points (resp. δ-cluster points). A point
is called a θ-cluster point (resp. a δ-cluster point) of a subset
if for every open neighborhood
of
in
the intersection
is not empty (resp.
is not empty).[13]
Using the fact that
and
whenever two subsets satisfy
the following may be deduced:
- Every α-open subset is semi-open, semi-preopen, preopen, and b-open.
- Every b-open set is semi-preopen (i.e. β-open).
- Every preopen set is b-open and semi-preopen.
- Every semi-open set is b-open and semi-preopen.
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]
See also
- Almost open map – Map that satisfies a condition similar to that of being an open map.
- Base (topology) – Collection of open sets used to define a topology
- Clopen set – Subset which is both open and closed
- Closed set – Complement of an open subset
- Domain (mathematical analysis) – Connected open subset of a topological space
- Local homeomorphism – Mathematical function revertible near each point
- Open map – A function that sends open (resp. closed) subsets to open (resp. closed) subsets
- Subbase – Collection of subsets that generate a topology
Notes
References
Bibliography
- Hart, Klaas (2004). Encyclopedia of general topology. Amsterdam Boston: Elsevier/North-Holland. ISBN 0-444-50355-2. OCLC 162131277.
- Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2004). Encyclopedia of general topology. Elsevier. ISBN 978-0-444-50355-8.
- Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.
External links
- "Open set", Encyclopedia of Mathematics, EMS Press, 2001 [1994]