In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
,
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, and related branches of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a closed set is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
whose
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class ...
is an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
.
In a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
, a closed set can be defined as a set which contains all its
limit point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
s. In a
complete metric space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the boun ...
, a closed set is a set which is
closed under the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
operation.
This should not be confused with a
closed manifold
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only connected one-dimensional example ...
.
Equivalent definitions
By definition, a subset
of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
is called if its complement
is an open subset of
; that is, if
A set is closed in
if and only if it is equal to its
closure in
Equivalently, a set is closed if and only if it contains all of its
limit point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
s. Yet another equivalent definition is that a set is closed if and only if it contains all of its
boundary points.
Every subset
is always contained in its
(topological) closure in
which is denoted by
that is, if
then
Moreover,
is a closed subset of
if and only if
An alternative characterization of closed sets is available via
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s and
nets. A subset
of a topological space
is closed in
if and only if every
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
of every net of elements of
also belongs to
In a
first-countable space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
(such as a metric space), it is enough to consider only convergent
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s, instead of all nets. One value of this characterization is that it may be used as a definition in the context of
convergence space
In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a that satisfies certain properties relating elements of ''X'' with the family of filters on ''X''. Convergence spaces generalize ...
s, which are more general than topological spaces. Notice that this characterization also depends on the surrounding space
because whether or not a sequence or net converges in
depends on what points are present in
A point
in
is said to be a subset
if
(or equivalently, if
belongs to the closure of
in the
topological subspace
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
meaning
where
is endowed with the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
induced on it by
[In particular, whether or not is close to depends only on the subspace and not on the whole surrounding space (e.g. or any other space containing as a topological subspace).]).
Because the closure of
in
is thus the set of all points in
that are close to
this terminology allows for a plain English description of closed subsets:
:a subset is closed if and only if it contains every point that is close to it.
In terms of net convergence, a point
is close to a subset
if and only if there exists some net (valued) in
that converges to
If
is a
topological subspace
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
of some other topological space
in which case
is called a of
then there exist some point in
that is close to
(although not an element of
), which is how it is possible for a subset
to be closed in
but to be closed in the "larger" surrounding super-space
If
and if
is topological super-space of
then
is always a (potentially proper) subset of
which denotes the closure of
in
indeed, even if
is a closed subset of
(which happens if and only if
), it is nevertheless still possible for
to be a proper subset of
However,
is a closed subset of
if and only if
for some (or equivalently, for every) topological super-space
of
Closed sets can also be used to characterize
continuous functions
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
: a map
is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
if and only if
for every subset
; this can be reworded in
plain English
Plain English (or layman's terms) are groups of words that are to be clear and easy to know. It usually avoids the use of rare words and uncommon euphemisms to explain the subject. Plain English wording is intended to be suitable for almost anyone, ...
as:
is continuous if and only if for every subset
maps points that are close to
to points that are close to
Similarly,
is continuous at a fixed given point
if and only if whenever
is close to a subset
then
is close to
More about closed sets
The notion of closed set is defined above in terms of
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
s, a concept that makes sense for
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s, as well as for other spaces that carry topological structures, such as
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s,
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s,
uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
s, and
gauge space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unif ...
s.
Whether a set is closed depends on the space in which it is embedded. However, the
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
s are "
absolutely closed", in the sense that, if you embed a compact Hausdorff space
in an arbitrary Hausdorff space
then
will always be a closed subset of
; the "surrounding space" does not matter here.
Stone–Čech compactification In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Stone ...
, a process that turns a
completely regular
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
Hausdorff space into a compact Hausdorff space, may be described as adjoining limits of certain nonconvergent nets to the space.
Furthermore, every closed subset of a compact space is compact, and every compact subspace of a Hausdorff space is closed.
Closed sets also give a useful characterization of compactness: a topological space
is compact if and only if every collection of nonempty closed subsets of
with empty intersection admits a finite subcollection with empty intersection.
A topological space
is
disconnected if there exist disjoint, nonempty, open subsets
and
of
whose union is
Furthermore,
is
totally disconnected
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
if it has an
open basis
In mathematics, a base (or basis) for the Topology (structure), topology of a topological space is a Family of sets, family \mathcal of Open set, open subsets of such that every open set of the topology is equal to the set union, union of some S ...
consisting of closed sets.
Properties
A closed set contains its own
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
. In other words, if you are "outside" a closed set, you may move a small amount in any direction and still stay outside the set. Note that this is also true if the boundary is the empty set, e.g. in the metric space of rational numbers, for the set of numbers of which the square is less than
* Any
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
of any family of closed sets is closed (this includes intersections of infinitely many closed sets)
* The
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of closed sets is closed.
* The
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
is closed.
* The whole set is closed.
In fact, if given a set
and a collection
of subsets of
such that the elements of
have the properties listed above, then there exists a unique topology
on
such that the closed subsets of
are exactly those sets that belong to
The intersection property also allows one to define the
closure of a set
in a space
which is defined as the smallest closed subset of
that is a
superset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of
Specifically, the closure of
can be constructed as the intersection of all of these closed supersets.
Sets that can be constructed as the union of
countably
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
many closed sets are denoted
Fσ sets. These sets need not be closed.
Examples
* The closed
interval in the real numbers.
* Some sets are both open and closed and are called clopen sets.
* The Line (geometry)#Ray">ray
[1, +\infty) is closed.
* The Cantor set is an unusual closed set in the sense that it consists entirely of boundary points and is nowhere dense.
* Singleton points (and thus finite sets) are closed in T1 space, T
1 spaces and
Hausdorff spaces
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the man ...
.
* The set of
integers
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
\Z is an infinite and unbounded closed set in the real numbers.
* If
f : X \to Y is a function between topological spaces then
f is continuous if and only if preimages of closed sets in
Y are closed in
X.
See also
*
*
*
*
*
*
*
Notes
References
*
*
*
*
{{DEFAULTSORT:Closed Set
General topology