TheInfoList

OR:

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 words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closin ...
, 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-clas ...
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 su ...
. 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 poi ...
, 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 conta ...
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 bo ...
, a closed set is a set which is closed under the limit 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 $A$ 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 poi ...
$\left(X, \tau\right)$ is called if its complement $X \setminus A$ is an open subset of $\left(X, \tau\right)$; that is, if $X \setminus A \in \tau.$ A set is closed in $X$ if and only if it is equal to its closure in $X.$ 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 conta ...
s. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. Every subset $A \subseteq X$ is always contained in its (topological) closure in $X,$ which is denoted by $\operatorname_X A;$ that is, if $A \subseteq X$ then $A \subseteq \operatorname_X A.$ Moreover, $A$ is a closed subset of $X$ if and only if $A = \operatorname_X A.$ 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 cal ...
s and nets. A subset $A$ of a topological space $X$ is closed in $X$ if and only if every limit of every net of elements of $A$ also belongs to $A.$ In a first-countable space (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 cal ...
s, instead of all nets. One value of this characterization is that it may be used as a definition in the context of convergence spaces, which are more general than topological spaces. Notice that this characterization also depends on the surrounding space $X,$ because whether or not a sequence or net converges in $X$ depends on what points are present in $X.$ A point $x$ in $X$ is said to be a subset $A \subseteq X$ if $x \in \operatorname_X A$ (or equivalently, if $x$ belongs to the closure of $A$ 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 ...
$A \cup \,$ meaning $x \in \operatorname_ A$ where $A \cup \$ 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 $X$In particular, whether or not $x$ is close to $A$ depends only on the subspace $A \cup \$ and not on the whole surrounding space (e.g. $X,$ or any other space containing $A \cup \$ as a topological subspace).). Because the closure of $A$ in $X$ is thus the set of all points in $X$ that are close to $A,$ 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 $x \in X$ is close to a subset $A$ if and only if there exists some net (valued) in $A$ that converges to $x.$ If $X$ 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 $Y,$ in which case $Y$ is called a of $X,$ then there exist some point in $Y \setminus X$ that is close to $A$ (although not an element of $X$), which is how it is possible for a subset $A \subseteq X$ to be closed in $X$ but to be closed in the "larger" surrounding super-space $Y.$ If $A \subseteq X$ and if $Y$ is topological super-space of $X$ then $A$ is always a (potentially proper) subset of $\operatorname_Y A,$ which denotes the closure of $A$ in $Y;$ indeed, even if $A$ is a closed subset of $X$ (which happens if and only if $A = \operatorname_X A$), it is nevertheless still possible for $A$ to be a proper subset of $\operatorname_Y A.$ However, $A$ is a closed subset of $X$ if and only if $A = X \cap \operatorname_Y A$ for some (or equivalently, for every) topological super-space $Y$ of $X.$ Closed sets can also be used to characterize continuous functions: a map $f : X \to Y$ 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 $f\left\left( \operatorname_X A \right\right) \subseteq \operatorname_Y \left(f\left(A\right)\right)$ for every subset $A \subseteq X$; 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: $f$ is continuous if and only if for every subset $A \subseteq X,$ $f$ maps points that are close to $A$ to points that are close to $f\left(A\right).$ Similarly, $f$ is continuous at a fixed given point $x \in X$ if and only if whenever $x$ is close to a subset $A \subseteq X,$ then $f\left(x\right)$ is close to $f\left(A\right).$

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 su ...
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 poi ...
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 set ...
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 un ...
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 un ...
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 Brit ...
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 $D$ in an arbitrary Hausdorff space $X,$ then $D$ will always be a closed subset of $X$; 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 St ...
, a process that turns a completely regular 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 $X$ is compact if and only if every collection of nonempty closed subsets of $X$ with empty intersection admits a finite subcollection with empty intersection. A topological space $X$ is disconnected if there exist disjoint, nonempty, open subsets $A$ and $B$ of $X$ whose union is $X.$ Furthermore, $X$ is totally disconnected if it has an open basis 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), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
. 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 $2.$ * 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, thei ...
of any family of closed sets is closed (this includes intersections of infinitely many closed sets) * The union 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 oth ...
is closed. * The whole set is closed. In fact, if given a set $X$ and a collection $\mathbb \neq \varnothing$ of subsets of $X$ such that the elements of $\mathbb$ have the properties listed above, then there exists a unique topology $\tau$ on $X$ such that the closed subsets of $\left(X, \tau\right)$ are exactly those sets that belong to $\mathbb.$ The intersection property also allows one to define the closure of a set $A$ in a space $X,$ which is defined as the smallest closed subset of $X$ 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 o ...
of $A.$ Specifically, the closure of $X$ can be constructed as the intersection of all of these closed supersets. Sets that can be constructed as the union of countably many closed sets are denoted Fσ sets. These sets need not be closed.

# Examples

* The closed interval
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s is closed. (See for an explanation of the bracket and parenthesis set notation.) * The
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysi ...

, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> is closed in the metric space of real numbers, and the set 
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\cap \Q of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s between $0$ and $1$ (inclusive) is closed in the space of rational numbers, but 
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\cap \Q is not closed in the real numbers. * Some sets are neither open nor closed, for instance the half-open interval _in_the_real_numbers. *_Some_sets_are_both_open_and_closed_and_are_called_clopen_sets. *_The_Line_(geometry)#Ray.html" "title="clopen_sets.html" ;"title=", 1) in the real numbers. * Some sets are both open and closed and are called clopen sets">, 1) in the real numbers. * Some sets are both open and closed and are called clopen sets. * The Line (geometry)#Ray">ray Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray ( ...
$\left[1, +\infty\right)$ 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, T1 spaces and Hausdorff spaces. * 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 languag ...
$\Z$ 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 $X.$