Closed set

Ingeometry,topology,and related branches ofmathematics,aclosed setis asetwhosecomplementis anopen set.^[1]^[2]In atopological space,a closed set can be defined as a set which contains all itslimit points.In acomplete metric space,a closed set is a set which isclosedunder thelimitoperation. This should not be confused with aclosed manifold.

Equivalent definitions[edit]

By definition, a subset $A$ of atopological space $(X,\tau )$ is calledclosedif its complement $X\setminus A$ is an open subset of $(X,\tau )$ ;that is, if $X\setminus A\in \tau.$ A set is closed in $X$ if and only if it is equal to itsclosurein $X.$ Equivalently, a set is closed if and only if it contains all of itslimit points.Yet another equivalent definition is that a set is closed if and only if it contains all of itsboundary points. Every subset $A\subseteq X$ is always contained in its(topological) closurein $X,$ which is denoted by $\operatorname {cl} _{X}A;$ that is, if $A\subseteq X$ then $A\subseteq \operatorname {cl} _{X}A.$ Moreover, $A$ is a closed subset of $X$ if and only if $A=\operatorname {cl} _{X}A.$

An alternative characterization of closed sets is available viasequencesandnets.A subset $A$ of a topological space $X$ is closed in $X$ if and only if everylimitof every net of elements of $A$ also belongs to $A.$ In afirst-countable space(such as a metric space), it is enough to consider only convergentsequences,instead of all nets. One value of this characterization is that it may be used as a definition in the context ofconvergence 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 beclose toa subset $A\subseteq X$ if $x\in \operatorname {cl} _{X}A$ (or equivalently, if $x$ belongs to the closure of $A$ in thetopological subspace $A\cup \{x\},$ meaning $x\in \operatorname {cl} _{A\cup \{x\}}A$ where $A\cup \{x\}$ is endowed with thesubspace topologyinduced on it by $X$ ^{[note 1]}). 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 atopological subspaceof some other topological space $Y,$ in which case $Y$ is called atopological super-spaceof $X,$ then theremightexist 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 tonotbe closed in the "larger" surrounding super-space $Y.$ If $A\subseteq X$ and if $Y$ isanytopological super-space of $X$ then $A$ is always a (potentially proper) subset of $\operatorname {cl} _{Y}A,$ which denotes the closure of $A$ in $Y;$ indeed, even if $A$ is a closed subset of $X$ (which happens if and only if $A=\operatorname {cl} _{X}A$ ), it is nevertheless still possible for $A$ to be a proper subset of $\operatorname {cl} _{Y}A.$ However, $A$ is a closed subset of $X$ if and only if $A=X\cap \operatorname {cl} _{Y}A$ for some (or equivalently, for every) topological super-space $Y$ of $X.$

Closed sets can also be used to characterizecontinuous functions:a map $f:X\to Y$ iscontinuousif and only if $f\left(\operatorname {cl} _{X}A\right)\subseteq \operatorname {cl} _{Y}(f(A))$ for every subset $A\subseteq X$ ;this can be reworded inplain Englishas: $f$ 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 $f(A).$ 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 $f(x)$ is close to $f(A).$

More about closed sets[edit]

The notion of closed set is defined above in terms ofopen sets,a concept that makes sense fortopological spaces,as well as for other spaces that carry topological structures, such asmetric spaces,differentiable manifolds,uniform spaces,andgauge spaces.

Whether a set is closed depends on the space in which it is embedded. However, thecompact Hausdorff spacesare "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,a process that turns acompletely regularHausdorff 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 $X$ isdisconnectedif there exist disjoint, nonempty, open subsets $A$ and $B$ of $X$ whose union is $X.$ Furthermore, $X$ istotally disconnectedif it has anopen basisconsisting of closed sets.

Properties[edit]

A closed set contains its ownboundary.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 $2.$

Anyintersectionof any family of closed sets is closed (this includes intersections of infinitely many closed sets)
Theunionoffinitelymanyclosed sets is closed.
Theempty setis closed.
The whole set is closed.

In fact, if given a set $X$ and a collection $\mathbb {F} \neq \varnothing$ of subsets of $X$ such that the elements of $\mathbb {F}$ have the properties listed above, then there exists a unique topology $\tau$ on $X$ such that the closed subsets of $(X,\tau )$ are exactly those sets that belong to $\mathbb {F}.$ The intersection property also allows one to define theclosureof a set $A$ in a space $X,$ which is defined as the smallest closed subset of $X$ that is asupersetof $A.$ 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 ofcountablymany closed sets are denotedF_σsets. These sets need not be closed.

Examples[edit]

The closedinterval $[a,b]$ ofreal numbersis closed. (SeeInterval (mathematics)for an explanation of the bracket and parenthesis set notation.)
Theunit interval $[0,1]$ is closed in the metric space of real numbers, and the set $[0,1]\cap \mathbb {Q}$ ofrational numbersbetween $0$ and $1$ (inclusive) is closed in the space of rational numbers, but $[0,1]\cap \mathbb {Q}$ is not closed in the real numbers.
Some sets are neither open nor closed, for instance the half-openinterval $[0,1)$ in the real numbers.
Some sets are both open and closed and are calledclopen sets.
Theray $[1,+\infty )$ is closed.
TheCantor setis 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 inT₁spacesandHausdorff spaces.
The set ofintegers $\mathbb {Z}$ 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 $X.$

Notes[edit]

^In particular, whether or not $x$ is close to $A$ depends only on thesubspace $A\cup \{x\}$ and not on the whole surrounding space (e.g. $X,$ or any other space containing $A\cup \{x\}$ as a topological subspace).

References[edit]

^Rudin, Walter(1976).Principles of Mathematical Analysis.McGraw-Hill.ISBN 0-07-054235-X.
^Munkres, James R.(2000).Topology(2nd ed.).Prentice Hall.ISBN 0-13-181629-2.

Dolecki, Szymon;Mynard, Frederic (2016).Convergence Foundations Of Topology.New Jersey: World Scientific Publishing Company.ISBN 978-981-4571-52-4.OCLC 945169917.
Dugundji, James(1966).Topology.Boston: Allyn and Bacon.ISBN 978-0-697-06889-7.OCLC 395340485.
Schechter, Eric(1996).Handbook of Analysis and Its Foundations.San Diego, CA: Academic Press.ISBN 978-0-12-622760-4.OCLC 175294365.
Willard, Stephen (2004) [1970].General Topology.Mineola, N.Y.:Dover Publications.ISBN 978-0-486-43479-7.OCLC 115240.

[3] In particular, whether or not $x$ is close to $A$ depends only on thesubspace $A\cup \{x\}$ and not on the whole surrounding space (e.g. $X,$ or any other space containing $A\cup \{x\}$ as a topological subspace).

[1] Rudin, Walter(1976).Principles of Mathematical Analysis.McGraw-Hill.ISBN 0-07-054235-X.

[2] Munkres, James R.(2000).Topology(2nd ed.).Prentice Hall.ISBN 0-13-181629-2.

[1]

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[note 1]

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