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Closed set

From Wikipedia, the free encyclopedia

Complement of an open subset

This article is about the complement of anopen set. For a set closed under an operation, seeclosure (mathematics). For other uses, seeClosed (disambiguation).

Ingeometry,topology, and related branches ofmathematics, aclosed set is aset whosecomplement is 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 isclosed under thelimit operation. This should not be confused withclosed manifold.

Sets that are both open and closed and are calledclopen sets.

Definition

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Given atopological space $(X,\tau )$ , the following statements are equivalent:

a set $A\subseteq X$ isclosed in $X . {\displaystyle X.}$
$A^{c}=X\setminus A$ is an open subset of $(X,\tau )$ ; that is, $A^{c}\in \tau .$
$A {\displaystyle A}$ is equal to itsclosure in $X . {\displaystyle X.}$
$A {\displaystyle A}$ contains all of itslimit points.
$A {\displaystyle A}$ contains all of itsboundary points.

An alternativecharacterization of closed sets is available viasequences andnets. A subset $A {\displaystyle A}$ of a topological space $X {\displaystyle X}$ is closed in $X {\displaystyle X}$ if and only if everylimit of every net of elements of $A {\displaystyle A}$ also belongs to $A . {\displaystyle A.}$ 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 $X, {\displaystyle X,}$ because whether or not a sequence or net converges in $X {\displaystyle X}$ depends on what points are present in $X . {\displaystyle X.}$ A point $x {\displaystyle x}$ in $X {\displaystyle X}$ is said to beclose to a subset $A\subseteq X$ if $x\in \operatorname {cl} _{X}A$ (or equivalently, if $x {\displaystyle x}$ belongs to the closure of $A {\displaystyle A}$ in thetopological subspace $A\cup \{x\},$ meaning $x\in \operatorname {cl} _{A\cup \{x\}}A$ where $A\cup \{x\}$ is endowed with thesubspace topology induced on it by $X {\displaystyle X}$ ^{[note 1]}). Because the closure of $A {\displaystyle A}$ in $X {\displaystyle X}$ is thus the set of all points in $X {\displaystyle X}$ that are close to $A, {\displaystyle A,}$ 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 $A {\displaystyle A}$ if and only if there exists some net (valued) in $A {\displaystyle A}$ that converges to $x . {\displaystyle x.}$ If $X {\displaystyle X}$ is atopological subspace of some other topological space $Y, {\displaystyle Y,}$ in which case $Y {\displaystyle Y}$ is called atopological super-space of $X, {\displaystyle X,}$ then theremight exist some point in $Y\setminus X$ that is close to $A {\displaystyle A}$ (although not an element of $X {\displaystyle X}$ ), which is how it is possible for a subset $A\subseteq X$ to be closed in $X {\displaystyle X}$ but tonot be closed in the "larger" surrounding super-space $Y . {\displaystyle Y.}$ If $A\subseteq X$ and if $Y {\displaystyle Y}$ isany topological super-space of $X {\displaystyle X}$ then $A {\displaystyle A}$ is always a (potentially proper) subset of $\operatorname {cl} _{Y}A,$ which denotes the closure of $A {\displaystyle A}$ in $Y; {\displaystyle Y;}$ indeed, even if $A {\displaystyle A}$ is a closed subset of $X {\displaystyle X}$ (which happens if and only if $A=\operatorname {cl} _{X}A$ ), it is nevertheless still possible for $A {\displaystyle A}$ to be a proper subset of $\operatorname {cl} _{Y}A.$ However, $A {\displaystyle A}$ is a closed subset of $X {\displaystyle X}$ if and only if $A=X\cap \operatorname {cl} _{Y}A$ for some (or equivalently, for every) topological super-space $Y {\displaystyle Y}$ of $X . {\displaystyle X.}$

Closed sets can also be used to characterizecontinuous functions: a map $f:X\to Y$ iscontinuous if 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 English as: $f {\displaystyle f}$ is continuous if and only if for every subset $A\subseteq X,$ $f {\displaystyle f}$ maps points that are close to $A {\displaystyle A}$ to points that are close to $f(A).$ Similarly, $f {\displaystyle f}$ is continuous at a fixed given point $x\in X$ if and only if whenever $x {\displaystyle x}$ is close to a subset $A\subseteq X,$ then $f(x)$ is close to $f(A).$

More about closed sets

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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 spaces are "absolutely closed", in the sense that, if you embed a compact Hausdorff space $D {\displaystyle D}$ in an arbitrary Hausdorff space $X, {\displaystyle X,}$ then $D {\displaystyle D}$ will always be a closed subset of $X {\displaystyle X}$ ; the "surrounding space" does not matter here.Stone–Čech compactification, a process that turns acompletely 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 $X {\displaystyle X}$ is compact if and only if every collection of nonempty closed subsets of $X {\displaystyle X}$ with empty intersection admits a finite subcollection with empty intersection.

A topological space $X {\displaystyle X}$ isdisconnected if there exist disjoint, nonempty, open subsets $A {\displaystyle A}$ and $B {\displaystyle B}$ of $X {\displaystyle X}$ whose union is $X . {\displaystyle X.}$ Furthermore, $X {\displaystyle X}$ istotally disconnected if it has anopen basis consisting of closed sets.

Properties

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Examples

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The closedinterval $[a,b]$ ofreal numbers is 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 numbers between $0 {\displaystyle 0}$ and $1 {\displaystyle 1}$ (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.
Theray $[1,+\infty )$ is closed.
TheCantor 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 inT₁ spaces andHausdorff 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 $f {\displaystyle f}$ is continuous if and only if preimages of closed sets in $Y {\displaystyle Y}$ are closed in $X . {\displaystyle X.}$

Notes

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

Citations

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^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.

References

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Dolecki, Szymon; Mynard, Frédéric (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.

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