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Closed set
 
 
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In topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
 and related branches of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a closed set is a set whose complement
Complement (set theory)

In discrete mathematics and predominantly in set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation to another....
 is open
Open set

In metric topology and related fields of mathematics, a Set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U....
.
topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
, a set is closed if and only if it coincides with its closure. Equivalently, a set is closed if and only if it contains all of its limit point
Limit point

In mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S other than x itself....
s.

This is not to be confused with a closed manifold
Closed manifold

In mathematics, a closed manifold is a type of topological space, namely a compact space manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
.

osed set contains its own boundary
Boundary (topology)

In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S....
. In other words, if you are "outside" a closed set and you "wiggle" a little bit, you will stay outside the set.






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In topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
 and related branches of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a closed set is a set whose complement
Complement (set theory)

In discrete mathematics and predominantly in set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation to another....
 is open
Open set

In metric topology and related fields of mathematics, a Set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U....
.

Equivalent definitions of a closed set

In a topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
, a set is closed if and only if it coincides with its closure. Equivalently, a set is closed if and only if it contains all of its limit point
Limit point

In mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S other than x itself....
s.

This is not to be confused with a closed manifold
Closed manifold

In mathematics, a closed manifold is a type of topological space, namely a compact space manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
.

Properties of closed sets

A closed set contains its own boundary
Boundary (topology)

In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S....
. In other words, if you are "outside" a closed set and you "wiggle" a little bit, you will 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
Intersection (set theory)

In mathematics, the intersection of two Set A and B is the set that contains all elements of A that also belong to B , but no other elements....
 of arbitrarily many closed sets is closed, and any union
Union (set theory)

In set theory, the term Union refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets....
 of finitely
Finite set

In mathematics, finite set is a Set that has a finite number of element . For example,is a finite set with five elements. The number of elements of a finite set is a natural number , and is called the cardinality of the set....
 many closed sets is closed. In particular, the empty set
Empty set

In mathematics, and more specifically set theory, the empty set is the unique Set having no members. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced....
 and the whole space are closed. In fact, given a set X and a collection F of subsets of X that has these properties, then F will be the collection of closed sets for a unique topology on X. The intersection property also allows one to define the closure
Closure (topology)

In mathematics, the closure of a set S consists of all Topology glossary#Ps which are intuitively "close to S". A point which is in the closure of S is a adherent point of S....
 of a set A in a space X, which is defined as the smallest closed subset of X that is a superset
SuperSet

SuperSet Software was a group founded by friends and former Eyring Research Institute co-workers Drew Major, Dale Neibaur, Kyle Powell and later joined by Mark Hurst....
 of A. Specifically, the closure of A 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 Fs
F-sigma set

In mathematics, an Fs set is a countable union of closed sets. The notation originated in France with F for wikt:ferm?#French and s for wikt:somme#French ....
 sets. These sets need not be closed.

Examples of closed sets

  • The closed interval
    Interval

    Interval may refer to:* Interval , a range of numbers * Interval measurements or interval variables in statistics is a level of measurement* Interval , the relationship between two notes...
     [a,b] of real number
    Real number

    In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
    s is closed. (See intervals for an explanation of the bracket and parenthesis set notation.)
  • The unit interval
    Unit interval

    In mathematics, the unit interval is the interval [0,1], that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1....
     [0,1] is closed in the metric space real numbers, and the set [0,1] n Q of rational number
    Rational number

    In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
    s between 0 and 1 (inclusive) is closed in the space of rational numbers, but [0,1] n Q is not closed in the real numbers.
  • Some sets are neither open nor closed, for instance the half-open interval
    Interval (mathematics)

    In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set....
     [0,1) in the real numbers.
  • Some sets are both open and closed and are called clopen sets.


More about closed sets

In point set topology a set is closed if it contains all its boundary
Boundary (topology)

In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S....
 points.

The notion of closed set is defined above in terms of open set
Open set

In metric topology and related fields of mathematics, a Set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U....
s, a concept that makes sense for topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
s, as well as for other spaces that carry topological structures, such as metric space
Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
s, differentiable manifold
Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to Euclidean space to allow one to do calculus. This article deals with differentiability in different contexts including: smooth function, k times differentiable, and holomorphic function....
s, uniform space
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 which is used to define uniform property such as complete space, uniform continuity and uniform convergence....
s, and gauge spaces.

An alternative characterization of closed sets is available via sequence
Sequence

In mathematics, a sequence is an ordered list of objects . Like a Set , it contains Element , and the number of terms is called the length of the sequence....
s and net
Net (mathematics)

In topology and related areas of mathematics a net or Moore-Smith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces....
s. A subset A of a topological space X is closed in X if and only if every limit
Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the behavior of a Function as its argument or input either "gets close" to some point, or as the argument becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely....
 of every net of elements of A also belongs to A. In a first-countable space
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 system ....
 (such as a metric space), it is enough to consider only convergent sequence
Sequence

In mathematics, a sequence is an ordered list of objects . Like a Set , it contains Element , and the number of terms is called the length of the sequence....
s, instead of all nets. One value of this characterisation 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 characterisation 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.

We have seen twice that whether a set is closed is relative depends on the space in which it is embedded. However, the compact
Compact space

In mathematics, a topological space is called compact if each of its open covers has a finite set subcover.Note: Some authors such as Nicolas Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff spaces and "quasi-compact"....
 Hausdorff space
Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhood ....
s are "absolutely closed" in a certain sense.To be precise, if you embed a compact Hausdorff space K in an arbitrary Hausdorff space X, then K will always be a closed subset of X; the "surrounding space" does not matter here. In fact, this property characterizes the compact Hausdorff spaces. Stone-Cech compactification, 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.