, open sets are a generalization
of open interval
s in the real line. In a metric space
—that is, when a distance is defined—open sets are the sets
that, with every point , contain all points that are sufficiently near to (that is all points whose distance to is less than some value depending on ).
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
, and compactness
, which were originally defined by means of a distance.
The most common case of a topology without any distance is given by manifold
s, which are topological spaces that, ''near'' each point, resemble an open set of an 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 subset
s, 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 space
In the set of all real number
s, 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
(−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.
of the Euclidean -space
is ''open'' if, for every point in
, there exists
a positive real number (depending on ) 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
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
satisfying , ''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.
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
are open sets)
(any union of open sets is an open set)
(any finite intersection of open sets is an open set)
Infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form
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 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
In 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 topological space. To see why
is clopen, begin by recalling that the sets
are, by definition, always open subsets (of
). Also by definition, a subset
is called 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 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
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 belong to the Euclidean topology since it is not a union of [[Interval (mathematics)#Including or excluding endpoints|open intervals of the form
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 non-empty proper subset
is an open subset or else a closed subset, but never both; that is, if
) then of the following two statements is true: either (1)
or else, (2)
Said differently, subset is open or closed but the subsets that are both (i.e. that are clopen) are
Regular open sets
of a topological space
is called a if
or equivalently, if
) denotes the topological boundary
A topological space for which there exists a base
consisting of regular open sets is called a .
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
is called a if
or equivalently, if
Every regular open set (resp. regular closed set) is an open subset (resp. is a closed subset) although in general,
[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 ]
the converses are true.
of any number of open sets, or infinitely many open sets, is open.
of a finite number of open sets is open.
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.
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
''A'' of a topological space ''X'' contains a (possibly empty) open set; the maximum (ordered under inclusion) such open set is called the interior
It can be constructed by taking the union of all the open sets contained in ''A''.
between two topological spaces
is if the preimage
of every open set in
is open in
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.
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 number
s, 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 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 positive ''a'' such that all points within distance ''a'' of ''x'' are in ''U'' (because ''U'' contains no non-rational numbers).
Generalizations of open sets
will be a topological space.
of a topological space
- if , and the complement of such a set is called .
- , , or if it satisfies any of the following equivalent conditions:
The complement of a preopen set is called .
- There exists subsets such that is open in is a dense subset of and
- There exists an open (in ) subset such that is a dense subset of
- if . The complement of a b-open set is called .
- or if it satisfies any of the following equivalent conditions:
The complement of a β-open set is called .
- is a regular closed subset of
- There exists a preopen subset of such that
- if it satisfies any of the following equivalent conditions:
The complement of a sequentially open set is called . A subset is sequentially closed in if and only if is equal to its , which by definition is the set consisting of all for which there exists a sequence in that converges to (in ).
- 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 in which by definition is the set
- and is said to have if there exists an open subset such that is a meager subset, where denotes the symmetric difference.
* 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 . [.]
- if . The complement in of a semi-open set is called a set.
* The (in ) of a subset denoted by is the intersection of all semi-closed subsets of that contain as a subset.
- if for each there exists some semiopen subset of such that
- (resp. ) if its complement in is a θ-closed (resp. ) set, where by definition, a subset of is called (resp. ) if it is equal to the set of all of its θ-cluster points (resp. δ-cluster points). A point is called a (resp. a ) of a subset if for every open neighborhood of in the intersection is not empty (resp. is not empty).
Using the fact that
whenever two subsets
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. 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. Preopen sets need not be semi-open and semi-open sets need not be preopen.
Arbitrary unions of preopen (resp. α-open, b-open, semi-preopen) sets are once again preopen (resp. α-open, b-open, semi-preopen). However, finite intersections of preopen sets need not be preopen. The set of all α-open subsets of a space
forms a topology on
that is finer
A topological space
if and only if every compact subspace
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 of every preopen subset is open.