# Closed set

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{{Short description|Complement of an open subset}}
{{About|the complement of an [open set](/source/open_set)|a set closed under an operation|closure (mathematics)|other uses|Closed (disambiguation)}}

In [topology](/source/topology), a branch of [mathematics](/source/mathematics), a '''closed set''' is a set that contains all of its boundary points. An example is the [closed interval](/source/closed_interval) <math>[a,b]</math>, which is closed in the real line because it includes both points <math>a</math> and <math>b</math> of its boundary. In general, a point is on the boundary if every neighborhood of it contains some points that belong to the set, and some points that do not. A set is thus closed if it is equal to its [closure](/source/closure_(topology)), the set obtained by adjoining all boundary points to it.

Closed sets are defined as subsets of [topological space](/source/topological_space)s. The topology of a space is usually described in terms of its [open set](/source/open_set)s, which determine what counts as a "neighborhood" of its points.  A set is closed if it is the [complement](/source/complement_(set_theory)) of an open set. In [metric space](/source/metric_space)s, a set is closed if and only if the limit of every [convergent sequence](/source/convergent_sequence) of elements in the set has limit in this set; thus a closed set is a set that includes all of its [limit point](/source/limit_point)s. Because the limits of convergent sequences do not escape a closed set, they are important in many areas of mathematics where limiting arguments are used.

Sets that are both open and closed are called [clopen sets](/source/clopen_sets).

== Definition ==
Given a [topological space](/source/topological_space) <math>(X, \tau)</math>, the following statements are equivalent:
# a set <math>A \subseteq X</math> is '''{{em|closed}}''' in <math>X.</math>
# <math>A^c = X \setminus A</math> is an open subset of <math>(X, \tau)</math>; that is, <math>A^{c} \in \tau.</math>
# <math>A</math> is equal to its [closure](/source/Closure_(topology)) in <math>X.</math>
# <math>A</math> contains all of its [limit point](/source/limit_point)s.
# <math>A</math> contains all of its [boundary points](/source/Boundary_(topology)). 

An alternative [characterization](/source/characterization_(mathematics)) of closed sets is available via [sequence](/source/sequence)s and [nets](/source/Net_(mathematics)).  A subset <math>A</math> of a topological space <math>X</math> is closed in <math>X</math> if and only if every [limit](/source/Limit_of_a_sequence) of every net of elements of <math>A</math> also belongs to <math>A.</math> In a [first-countable space](/source/first-countable_space) (such as a metric space), it is enough to consider only convergent [sequence](/source/sequence)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](/source/convergence_space)s, which are more general than topological spaces. Notice that this characterization also depends on the surrounding space <math>X,</math> because whether or not a sequence or net converges in <math>X</math> depends on what points are present in <math>X.</math> 
A point <math>x</math> in <math>X</math> is said to be {{em|close to}} a subset <math>A \subseteq X</math> if <math>x \in \operatorname{cl}_X A</math> (or equivalently, if <math>x</math> belongs to the closure of <math>A</math> in the [topological subspace](/source/topological_subspace) <math>A \cup \{ x \},</math> meaning <math>x \in \operatorname{cl}_{A \cup \{ x \}} A</math> where <math>A \cup \{ x \}</math> is endowed with the [subspace topology](/source/subspace_topology) induced on it by <math>X</math><ref group="note">In particular, whether or not <math>x</math> is close to <math>A</math> depends only on the [subspace](/source/Topological_subspace) <math>A \cup \{ x \}</math> and not on the whole surrounding space (e.g. <math>X,</math> or any other space containing <math>A \cup \{ x \}</math> as a topological subspace).</ref>).  
Because the closure of <math>A</math> in <math>X</math> is thus the set of all points in <math>X</math> that are close to <math>A,</math> this terminology allows for an intuitive 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 <math>x \in X</math> is close to a subset <math>A</math> if and only if there exists some net (valued) in <math>A</math> that converges to <math>x.</math> 
If <math>X</math> is a [topological subspace](/source/topological_subspace) of some other topological space <math>Y,</math> in which case <math>Y</math> is called a {{em|topological super-space}} of <math>X,</math> then there {{em|might}} exist some point in <math>Y \setminus X</math> that is close to <math>A</math> (although not an element of <math>X</math>), which is how it is possible for a subset <math>A \subseteq X</math> to be closed in <math>X</math> but to {{em|not}} be closed in the "larger" surrounding super-space <math>Y.</math> 
If <math>A \subseteq X</math> and if <math>Y</math> is {{em|any}} topological super-space of <math>X</math> then <math>A</math> is always a (potentially proper) subset of <math>\operatorname{cl}_Y A,</math> which denotes the closure of <math>A</math> in <math>Y;</math> indeed, even if <math>A</math> is a closed subset of <math>X</math> (which happens if and only if <math>A = \operatorname{cl}_X A</math>), it is nevertheless still possible for <math>A</math> to be a proper subset of <math>\operatorname{cl}_Y A.</math> However, <math>A</math> is a closed subset of <math>X</math> if and only if <math>A = X \cap \operatorname{cl}_Y A</math> for some (or equivalently, for every) topological super-space <math>Y</math> of <math>X.</math> 

Closed sets can also be used to characterize [continuous functions](/source/Continuous_map): a map <math>f : X \to Y</math> is [continuous](/source/Continuous_function) if and only if <math>f\left( \operatorname{cl}_X A \right) \subseteq \operatorname{cl}_Y (f(A))</math> for every subset <math>A \subseteq X</math>; this can be reworded intuitively as: <math>f</math> is continuous if and only if for every subset <math>A \subseteq X,</math> <math>f</math> maps points that are close to <math>A</math> to points that are close to <math>f(A).</math> Similarly, <math>f</math> is continuous at a fixed given point <math>x \in X</math> if and only if whenever <math>x</math> is close to a subset <math>A \subseteq X,</math> then <math>f(x)</math> is close to <math>f(A).</math>

== More about closed sets ==

The notion of closed set is defined above in terms of [open set](/source/open_set)s, a concept that makes sense for [topological space](/source/topological_space)s, as well as for other spaces that carry topological structures, such as [metric space](/source/metric_space)s, [differentiable manifold](/source/differentiable_manifold)s, [uniform space](/source/uniform_space)s, and [gauge space](/source/gauge_space)s. 

Whether a set is closed depends on the space in which it is embedded. However, the [compact](/source/Compact_space) [Hausdorff space](/source/Hausdorff_space)s are "[absolutely closed](/source/H-closed_space)", in the sense that, if you embed a compact Hausdorff space <math>D</math> in an arbitrary Hausdorff space <math>X,</math> then <math>D</math> will always be a closed subset of <math>X</math>; the "surrounding space" does not matter here. [Stone–Čech compactification](/source/Stone%E2%80%93%C4%8Cech_compactification), a process that turns a [completely regular](/source/Completely_regular_space) 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 <math>X</math> is compact if and only if every collection of nonempty closed subsets of <math>X</math> with empty intersection admits a finite subcollection with empty intersection.

A topological space <math>X</math> is [disconnected](/source/Disconnected_space) if there exist disjoint, nonempty, open subsets <math>A</math> and <math>B</math> of <math>X</math> whose union is <math>X.</math> Furthermore, <math>X</math> is [totally disconnected](/source/totally_disconnected) if it has an [open basis](/source/open_basis) consisting of closed sets.

== Properties ==
{{See also|Kuratowski closure axioms}}

A closed set contains its own [boundary](/source/Boundary_(topology)). 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. 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 <math>2.</math> 

* Any [intersection](/source/Intersection_(set_theory)) of any family of closed sets is closed (this includes intersections of infinitely many closed sets)
* The [union](/source/Union_(set_theory)) of {{em|[finitely](/source/Finite_set) many}} closed sets is closed.
* The [empty set](/source/empty_set) is closed.
* The whole set is closed.

In fact, if given a set <math>X</math> and a collection <math>\mathbb{F} \neq \varnothing</math> of subsets of <math>X</math> such that the elements of <math>\mathbb{F}</math> have the properties listed above, then there exists a unique topology <math>\tau</math> on <math>X</math> such that the closed subsets of <math>(X, \tau)</math> are exactly those sets that belong to <math>\mathbb{F}.</math> 
The intersection property also allows one to define the [closure](/source/Closure_(topology)) of a set <math>A</math> in a space <math>X,</math> which is defined as the smallest closed subset of <math>X</math> that is a [superset](/source/superset) of <math>A.</math>
Specifically, the closure of <math>X</math> can be constructed as the intersection of all of these closed supersets.

Sets that can be constructed as the union of [countably](/source/Countable_set) many closed sets are denoted '''[F<sub>σ</sub>](/source/F-sigma_set)''' sets.  These sets need not be closed.

== Examples ==
* The closed [interval](/source/Interval_(mathematics)) <math>[a, b]</math> of [real number](/source/real_number)s is closed. (See {{em|[Interval (mathematics)](/source/Interval_(mathematics))}} for an explanation of the bracket and parenthesis set notation.)
* The [unit interval](/source/unit_interval) <math>[0, 1]</math> is closed in the metric space of real numbers, and the set <math>[0, 1] \cap \Q</math> of [rational number](/source/rational_number)s between <math>0</math> and <math>1</math> (inclusive) is closed in the space of rational numbers, but <math>[0, 1] \cap \Q</math> is not closed in the real numbers.
* Some sets are neither open nor closed, for instance the half-open [interval](/source/Interval_(mathematics)) <math>[0, 1)</math> in the real numbers.
* In the finite complement topology on a set <math>X</math>, the closed sets are precisely the finite subsets of <math>X</math> together with <math>X</math> itself.
* In the discrete topology on a set <math>X</math>, every subset of <math>X</math> is closed.

* The [ray](/source/Line_(geometry)) <math>[1, +\infty)</math> is closed.
* The [Cantor set](/source/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 [T<sub>1</sub> spaces](/source/T1_space) and [Hausdorff spaces](/source/Hausdorff_spaces).
* The set of [integers](/source/integers) <math>\Z</math> is an infinite and unbounded closed set in the real numbers.
* If <math>f : X \to Y</math> is a function between topological spaces then <math>f</math> is continuous if and only if preimages of closed sets in <math>Y</math> are closed in <math>X.</math>
* Each [lower set](/source/Upper_set) of a [preorder](/source/Preorder) is closed in the [Alexandrov topology](/source/Alexandrov_topology) on the preorder.
* [Compact set](/source/Compact_set)s in a [Hausdorff space](/source/Hausdorff_space) are always closed. Compactness is a specialization of the idea of compactness that, in cases such as metric spaces, ensures that not only does a set contain all of its limits, but that ''every'' sequence has a subsequence with a limit in the set.

== Uses and importance ==
Closed sets are important throughout mathematics because they describe conditions that are preserved under limiting processes. In a metric space, for example, if a sequence of points in a closed set converges in the ambient space, then its limit remains in the set. Thus one can often prove that an object has a desired property by constructing it as a limit of objects that already have that property. Closed sets are therefore ubiquitous throughout [mathematical analysis](/source/mathematical_analysis), which involve limiting arguments throughout.

[Continuous maps](/source/continuous_function) provide one source of closed sets in many applications. A function between topological spaces is continuous if and only if the [inverse image](/source/inverse_image) of every closed set is closed. Consequently, solution sets of continuous equations are closed: if <math>f:X\to\mathbb R</math> is continuous, then the zero set
<math display="block">\{x\in X:f(x)=0\}=f^{-1}(\{0\})</math>
is closed. More generally, level sets and [constraint sets](/source/constrained_optimization) defined by continuous equalities are closed.

In [algebraic geometry](/source/algebraic_geometry), closed sets are used to encode systems of polynomial equations. In the [Zariski topology](/source/Zariski_topology) on affine space, the closed sets are the [algebraic set](/source/algebraic_set)s, that is, the common zero sets of collections of [polynomial](/source/polynomial)s. So in algebraic geometry, the closed sets, rather than the open sets, are often the primary objects of study.

In [functional analysis](/source/functional_analysis), closedness is used to control infinite-dimensional limiting processes. A linear subspace of a normed vector space need not be closed. When it is not closed, limits of convergent sequences of vectors in the subspace may leave the subspace. Closed subspaces of [Banach](/source/Banach_space) and [Hilbert space](/source/Hilbert_space)s are therefore especially important. Similarly, the [closed graph theorem](/source/closed_graph_theorem) characterizes continuity of certain linear operators between Banach spaces by the closedness of their [graphs](/source/graph_of_a_function).

== See also ==
* {{annotated link|Clopen set}}
* {{annotated link|Closed map}}
* {{annotated link|Closed region}}
* {{annotated link|Open set}}
* {{annotated link|Neighbourhood (mathematics)|Neighbourhood}}
* {{annotated link|Region (mathematics)}}
* {{annotated link|Regular closed set}}

== Notes ==
{{reflist|group=note}}

== Citations ==
{{reflist}}

== References ==
* {{Dolecki Mynard Convergence Foundations Of Topology}} <!-- {{sfn|Dolecki|Mynard|2016|p=}} -->
* {{Dugundji Topology}} <!-- {{sfn|Dugundji|1966|p=}} -->
* {{Schechter Handbook of Analysis and Its Foundations}} <!-- {{sfn|Schechter|1996|p=}} -->
* {{Willard General Topology}} <!-- {{sfn|Willard|2004|p=}} -->

{{Topology}}

{{DEFAULTSORT:Closed Set}}
Category:General topology

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Adapted from the Wikipedia article [Closed set](https://en.wikipedia.org/wiki/Closed_set) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Closed_set?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.