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Open set

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Basic subset of a topological space

Example: the bluecircle represents the set of points (x,y) satisfyingx² +y² =r². The reddisk represents the set of points (x,y) satisfyingx² +y² <r². The red set is an open set, the blue set is itsboundary set, and the union of the red and blue sets is aclosed set.

Inmathematics, anopen set is ageneralization of anopen interval in thereal line.

In ametric space (aset with adistance defined between every two points), an open set is a set that, with every pointP in it, contains all points of the metric space that are sufficiently near toP (that is, all points whose distance toP is less than some value depending onP).

More generally, an open set is a member of a givencollection ofsubsets of a given set, a collection that has the property of containing everyunion of its members, every finiteintersection of its members, theempty set, and the whole set itself. A set in which such a collection is given is called atopological space, and the collection is called atopology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example,every subset can be open (thediscrete topology), orno subset can be open except the space itself and the empty set (theindiscrete topology).^[1]

In practice, however, open sets are usually chosen to provide a notion of nearness that is similar to that of metric spaces, without having a notion of distance defined. In particular, a topology allows defining properties such ascontinuity,connectedness, andcompactness, which were originally defined by means of a distance.

The most common case of a topology without any distance is given bymanifolds, which are topological spaces that,near each point, resemble an open set of aEuclidean space, but on which no distance is defined in general. Less intuitive topologies are used in other branches of mathematics; for example, theZariski topology, which is fundamental inalgebraic geometry andscheme theory.

Motivation

Intuitively, an open set provides a method to distinguish twopoints. For example, if about one of two points in atopological space, there exists an open set not containing the other (distinct) point, the two points are referred to astopologically distinguishable. In this manner, one may speak of whether two points, or more generally twosubsets, of a topological space are "near" without concretely defining adistance. Therefore, topological spaces may be seen as a generalization of spaces equipped with a notion of distance, which are calledmetric spaces.

In the set of allreal numbers, one has the naturalEuclidean metric; that is, a function which measures the distance between two real numbers:d(x,y) = |x −y|. Therefore, given a real numberx, one can speak of the set of all points close to that real number; that is, withinε ofx. In essence, points within ε ofx approximatex to an accuracy of degreeε. Note thatε > 0 always but asε becomes smaller and smaller, one obtains points that approximatex to a higher and higher degree of accuracy. For example, ifx = 0 andε = 1, the points withinε ofx are precisely the points of theinterval (−1, 1); that is, the set of all real numbers between −1 and 1. However, withε = 0.5, the points withinε ofx are precisely the points of (−0.5, 0.5). Clearly, these points approximatex to a greater degree of accuracy than whenε = 1.

The previous discussion shows, for the casex = 0, that one may approximatex to higher and higher degrees of accuracy by definingε to be smaller and smaller. In particular, sets of the form (−ε,ε) give us a lot of information about points close tox = 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close tox. This innovative idea has far-reaching consequences; in particular, by defining different collections of sets containing 0 (distinct from the sets (−ε,ε)), one may find different results regarding the distance between 0 and other real numbers. For example, if we were to defineR as the only such set for "measuring distance", all points are close to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a member ofR. Thus, we find that in some sense, every real number is distance 0 away from 0. It may help in this case to think of the measure as being a binary condition: all things inR are equally close to 0, while any item that is not inR is not close to 0.

In general, one refers to the family of sets containing 0, used to approximate 0, as aneighborhood basis; a member of this neighborhood basis is referred to as an open set. In fact, one may generalize these notions to an arbitrary set (X); rather than just the real numbers. In this case, given a point (x) of that set, one may define a collection of sets "around" (that is, containing)x, used to approximatex. Of course, this collection would have to satisfy certain properties (known asaxioms) for otherwise we may not have a well-defined method to measure distance. For example, every point inX should approximatex tosome degree of accuracy. ThusX should be in this family. Once we begin to define "smaller" sets containingx, we tend to approximatex to a greater degree of accuracy. Bearing this in mind, one may define the remaining axioms that the family of sets aboutx is required to satisfy.

Definitions

Several definitions are given here, in an increasing order of technicality. Each one is a special case of the next one.

Euclidean space

A subset $U {\displaystyle U}$ of theEuclideann-spaceRⁿ isopen if, for every pointx in $U {\displaystyle U}$ ,there exists a positive real numberε (depending onx) such that any point inRⁿ whoseEuclidean distance fromx is smaller thanε belongs to $U {\displaystyle U}$ .^[2] Equivalently, a subset $U {\displaystyle U}$ ofRⁿ is open if every point in $U {\displaystyle U}$ is the center of anopen ball contained in $U . {\displaystyle U.}$

An example of a subset ofR that is not open is theclosed interval[0,1], since neither0 -ε nor1 +ε belongs to[0,1] for anyε > 0, no matter how small.

Metric space

A subsetU of ametric space(M,d) is calledopen if, for any pointx inU, there exists a real numberε > 0 such that any point $y\in M$ satisfyingd(x,y) <ε belongs toU. Equivalently,U is open if every point inU has a neighborhood contained inU.

This generalizes the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.

Topological space

Atopology $\tau$ on a setX is a set of subsets ofX with the properties below. Each member of $\tau$ is called anopen set.^[3]

$X\in \tau$ and $\varnothing \in \tau$
Any union of sets in $\tau$ belong to $\tau$ : if $\left\{U_{i}:i\in I\right\}\subseteq \tau$ then $\bigcup _{i\in I}U_{i}\in \tau$
Any finite intersection of sets in $\tau$ belong to $\tau$ : if $U_{1},\ldots ,U_{n}\in \tau$ then $U_{1}\cap \cdots \cap U_{n}\in \tau$

X together with $\tau$ is called atopological space.

Infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form $\left(-1/n,1/n\right),$ where $n {\displaystyle n}$ is a positive integer, is the set $\{0\}$ which is not open in the real line.

A metric space is a topological space, whose topology consists of the collection of all subsets that are unions of open balls. There are, however, topological spaces that are not metric spaces.

Properties

Theunion of any number of open sets, or infinitely many open sets, is open.^[4] Theintersection of a finite number of open sets is open.^[4]

Acomplement of an open set (relative to the space that the topology is defined on) is called aclosed set. A set may be both open and closed (aclopen set). Theempty set and the full space are examples of sets that are both open and closed.^[5]

A set can never been considered as open by itself. This notion is relative to a containing set and a specific topology on it.

Whether a set is open depends on thetopology under consideration. Having opted forgreater brevity over greater clarity, we refer to a setX endowed with a topology $\tau$ as "the topological spaceX" rather than "the topological space $(X,\tau )$ ", despite the fact that all the topological data is contained in $\tau .$ If there are two topologies on the same set, a setU that is open in the first topology might fail to be open in the second topology. For example, ifX is any topological space andY is any subset ofX, the setY can be given its own topology (called the 'subspace topology') defined by "a setU is open in the subspace topology onY if and only ifU is the intersection ofY with an open set from the original topology onX."^[6] This potentially introduces new open sets: ifV is open in the original topology onX, but $V\cap Y$ isn't open in the original topology onX, then $V\cap Y$ is open in the subspace topology onY.

As a concrete example of this, ifU is defined as the set of rational numbers in the interval $(0,1),$ thenU is an open subset of therational numbers, but not of thereal numbers. This is because when the surrounding space is the rational numbers, for every pointx inU, there exists a positive numbera such that allrational points within distancea ofx are also inU. On the other hand, when the surrounding space is the reals, then for every pointx inU there isno positivea such that allreal points within distancea ofx are inU (becauseU contains no non-rational numbers).

Uses

Open sets have a fundamental importance intopology. The concept is required to define and make sense oftopological space and other topological structures that deal with the notions of closeness and convergence for spaces such asmetric spaces anduniform spaces.

EverysubsetA of a topological spaceX contains a (possibly empty) open set; the maximum (ordered under inclusion) such open set is called theinterior ofA. It can be constructed by taking the union of all the open sets contained inA.^[7]

Afunction $f:X\to Y$ between two topological spaces $X {\displaystyle X}$ and $Y {\displaystyle Y}$ iscontinuous if thepreimage of every open set in $Y {\displaystyle Y}$ is open in $X . {\displaystyle X.}$ ^[8]The function $f:X\to Y$ is calledopen if theimage of every open set in $X {\displaystyle X}$ is open in $Y . {\displaystyle Y.}$

An open set on thereal line has the characteristic property that it is a countable union of disjoint open intervals.

Special types of open sets

Clopen sets and non-open and/or non-closed sets

A set might be open, closed, both, or neither. In particular, open and closed sets are not mutually exclusive, meaning that it is in general possible for a subset of a topological space to simultaneously be both an open subsetand a closed subset. Such subsets are known asclopen sets. Explicitly, a subset $S {\displaystyle S}$ of a topological space $(X,\tau )$ is calledclopen if both $S {\displaystyle S}$ and its complement $X\setminus S$ are open subsets of $(X,\tau )$ ; or equivalently, if $S\in \tau$ and $X\setminus S\in \tau .$

Inany topological space $(X,\tau ),$ the empty set $\varnothing$ and the set $X {\displaystyle X}$ itself are always clopen. These two sets are the most well-known examples of clopen subsets and they show that clopen subsets exist inevery topological space. To see, it suffices to remark that, by definition of a topology, $X {\displaystyle X}$ and $\varnothing$ are both open, and that they are also closed, since each is the complement of the other.

The open sets of the usualEuclidean topology of thereal line $\mathbb {R}$ are the empty set, theopen intervals and every union of open intervals.

The interval $I=(0,1)$ is open in $\mathbb {R}$ by definition of the Euclidean topology. It is not closed since its complement in $\mathbb {R}$ is $I^{\complement }=(-\infty ,0]\cup [1,\infty ),$ which is not open; indeed, an open interval contained in $I^{\complement }$ cannot contain1, and it follows that $I^{\complement }$ cannot be a union of open intervals. Hence, $I {\displaystyle I}$ is an example of a set that is open but not closed.
By a similar argument, the interval $J=[0,1]$ is a closed subset but not an open subset.
Finally, neither $K=[0,1)$ nor its complement $\mathbb {R} \setminus K=(-\infty ,0)\cup [1,\infty )$ are open (because they cannot be written as a union of open intervals); this means that $K {\displaystyle K}$ is neither open nor closed.

If a topological space $X {\displaystyle X}$ is endowed with thediscrete topology (so that by definition, every subset of $X {\displaystyle X}$ is open) then every subset of $X {\displaystyle X}$ is a clopen subset. For a more advanced example reminiscent of the discrete topology, suppose that ${\mathcal {U}}$ is anultrafilter on a non-empty set $X . {\displaystyle X.}$ Then the union $\tau :={\mathcal {U}}\cup \{\varnothing \}$ is a topology on $X {\displaystyle X}$ with the property thatevery non-empty proper subset $S {\displaystyle S}$ of $X {\displaystyle X}$ iseither an open subset or else a closed subset, but never both; that is, if $\varnothing \neq S\subsetneq X$ (where $S\neq X$ ) thenexactly one of the following two statements is true: either (1) $S\in \tau$ or else, (2) $X\setminus S\in \tau .$ Said differently,every subset is open or closed but theonly subsets that are both (i.e. that are clopen) are $\varnothing$ and $X . {\displaystyle X.}$

Regular open sets

A subset $S {\displaystyle S}$ of a topological space $X {\displaystyle X}$ is called aregular open set if $\operatorname {Int} \left({\overline {S}}\right)=S$ or equivalently, if $\operatorname {Bd} \left({\overline {S}}\right)=\operatorname {Bd} S$ , where $\operatorname {Bd} S$ , $\operatorname {Int} S$ , and ${\overline {S}}$ denote, respectively, the topologicalboundary,interior, andclosure of $S {\displaystyle S}$ in $X {\displaystyle X}$ . A topological space for which there exists abase consisting of regular open sets is called asemiregular space. A subset of $X {\displaystyle X}$ is a regular open set if and only if its complement in $X {\displaystyle X}$ is a regular closed set, where by definition a subset $S {\displaystyle S}$ of $X {\displaystyle X}$ is called aregular closed set if ${\overline {\operatorname {Int} S}}=S$ or equivalently, if $\operatorname {Bd} \left(\operatorname {Int} S\right)=\operatorname {Bd} S.$ Every regular open set (resp. regular closed set) is an open subset (resp. is a closed subset) although in general,^{[note 1]} the converses arenot true.

Generalizations of open sets

See also:Almost open map andGlossary of topology

Throughout, $(X,\tau )$ will be a topological space.

A subset $A\subseteq X$ of a topological space $X {\displaystyle X}$ is called:

α-open if $A~\subseteq ~\operatorname {int} _{X}\left(\operatorname {cl} _{X}\left(\operatorname {int} _{X}A\right)\right)$ , and the complement of such a set is calledα-closed.^[9]
preopen,nearly open, orlocallydense if it satisfies any of the following equivalent conditions:
1. $A~\subseteq ~\operatorname {int} _{X}\left(\operatorname {cl} _{X}A\right).$ ^[10]
2. There exists subsets $D,U\subseteq X$ such that $U {\displaystyle U}$ is open in $X, {\displaystyle X,}$ $D {\displaystyle D}$ is adense subset of $X, {\displaystyle X,}$ and $A=U\cap D.$ ^[10]
3. There exists an open (in $X {\displaystyle X}$ ) subset $U\subseteq X$ such that $A {\displaystyle A}$ is a dense subset of $U . {\displaystyle U.}$ ^[10]
The complement of a preopen set is calledpreclosed.
b-open if $A~\subseteq ~\operatorname {int} _{X}\left(\operatorname {cl} _{X}A\right)~\cup ~\operatorname {cl} _{X}\left(\operatorname {int} _{X}A\right)$ . The complement of a b-open set is calledb-closed.^[9]
β-open orsemi-preopen if it satisfies any of the following equivalent conditions:
1. $A~\subseteq ~\operatorname {cl} _{X}\left(\operatorname {int} _{X}\left(\operatorname {cl} _{X}A\right)\right)$ ^[9]
2. $\operatorname {cl} _{X}A$ is a regular closed subset of $X . {\displaystyle X.}$ ^[10]
3. There exists a preopen subset $U {\displaystyle U}$ of $X {\displaystyle X}$ such that $U\subseteq A\subseteq \operatorname {cl} _{X}U.$ ^[10]
The complement of a β-open set is calledβ-closed.
sequentially open if it satisfies any of the following equivalent conditions:
1. Whenever a sequence in $X {\displaystyle X}$ converges to some point of $A, {\displaystyle A,}$ then that sequence is eventually in $A . {\displaystyle A.}$ Explicitly, this means that if $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }$ is a sequence in $X {\displaystyle X}$ and if there exists some $a\in A$ is such that $x_{\bullet }\to x$ in $(X,\tau ),$ then $x_{\bullet }$ is eventually in $A {\displaystyle A}$ (that is, there exists some integer $i {\displaystyle i}$ such that if $j\geq i,$ then $x_{j}\in A$ ).
2. $A {\displaystyle A}$ is equal to itssequential interior in $X, {\displaystyle X,}$ which by definition is the set
  ${\begin{alignedat}{4}\operatorname {SeqInt} _{X}A:&=\{a\in A~:~{\text{ whenever a sequence in }}X{\text{ converges to }}a{\text{ in }}(X,\tau ),{\text{ then that sequence is eventually in }}A\}\\&=\{a\in A~:~{\text{ there does NOT exist a sequence in }}X\setminus A{\text{ that converges in }}(X,\tau ){\text{ to a point in }}A\}\\\end{alignedat}}$
The complement of a sequentially open set is calledsequentially closed. A subset $S\subseteq X$ is sequentially closed in $X {\displaystyle X}$ if and only if $S {\displaystyle S}$ is equal to itssequential closure, which by definition is the set $\operatorname {SeqCl} _{X}S$ consisting of all $x\in X$ for which there exists a sequence in $S {\displaystyle S}$ that converges to $x {\displaystyle x}$ (in $X {\displaystyle X}$ ).
almost open and is said to havethe Baire property if there exists an open subset $U\subseteq X$ such that $A\bigtriangleup U$ is ameager subset, where $\bigtriangleup$ denotes thesymmetric difference.^[11]
- The subset $A\subseteq X$ is said to havethe Baire property in the restricted sense if for every subset $E {\displaystyle E}$ of $X {\displaystyle X}$ the intersection $A\cap E$ has the Baire property relative to $E {\displaystyle E}$ .^[12]
semi-open if $A~\subseteq ~\operatorname {cl} _{X}\left(\operatorname {int} _{X}A\right)$ or, equivalently, $\operatorname {cl} _{X}A=\operatorname {cl} _{X}\left(\operatorname {int} _{X}A\right)$ . The complement in $X {\displaystyle X}$ of a semi-open set is called asemi-closed set.^[13]
- Thesemi-closure (in $X {\displaystyle X}$ ) of a subset $A\subseteq X,$ denoted by $\operatorname {sCl} _{X}A,$ is the intersection of all semi-closed subsets of $X {\displaystyle X}$ that contain $A {\displaystyle A}$ as a subset.^[13]
semi-θ-open if for each $x\in A$ there exists some semiopen subset $U {\displaystyle U}$ of $X {\displaystyle X}$ such that $x\in U\subseteq \operatorname {sCl} _{X}U\subseteq A.$ ^[13]
θ-open (resp.δ-open) if its complement in $X {\displaystyle X}$ is a θ-closed (resp.δ-closed) set, where by definition, a subset of $X {\displaystyle X}$ is calledθ-closed (resp.δ-closed) if it is equal to the set of all of its θ-cluster points (resp. δ-cluster points). A point $x\in X$ is called aθ-cluster point (resp. aδ-cluster point) of a subset $B\subseteq X$ if for every open neighborhood $U {\displaystyle U}$ of $x {\displaystyle x}$ in $X, {\displaystyle X,}$ the intersection $B\cap \operatorname {cl} _{X}U$ is not empty (resp. $B\cap \operatorname {int} _{X}\left(\operatorname {cl} _{X}U\right)$ is not empty).^[13]

Using the fact that

A~\subseteq ~\operatorname {cl} _{X}A~\subseteq ~\operatorname {cl} _{X}B

and

\operatorname {int} _{X}A~\subseteq ~\operatorname {int} _{X}B~\subseteq ~B

whenever two subsets $A,B\subseteq X$ satisfy $A\subseteq B,$ the following may be deduced:

Every α-open subset is semi-open, semi-preopen, preopen, and b-open.
Every b-open set is semi-preopen (i.e. β-open).
Every preopen set is b-open and semi-preopen.
Every semi-open set is b-open and semi-preopen.

Moreover, a subset is a regular open set if and only if it is preopen and semi-closed.^[10] The intersection of an α-open set and a semi-preopen (resp. semi-open, preopen, b-open) set is a semi-preopen (resp. semi-open, preopen, b-open) set.^[10] Preopen sets need not be semi-open and semi-open sets need not be preopen.^[10]

Arbitrary unions of preopen (resp. α-open, b-open, semi-preopen) sets are once again preopen (resp. α-open, b-open, semi-preopen).^[10] However, finite intersections of preopen sets need not be preopen.^[13] The set of all α-open subsets of a space $(X,\tau )$ forms a topology on $X {\displaystyle X}$ that isfiner than $\tau .$ ^[9]

A topological space $X {\displaystyle X}$ isHausdorff if and only if everycompact subspace of $X {\displaystyle X}$ is θ-closed.^[13] A space $X {\displaystyle X}$ istotally disconnected if and only if every regular closed subset is preopen or equivalently, if every semi-open subset is preopen. Moreover, the space is totally disconnected if and only if theclosure of every preopen subset is open.^[9]

See also

Almost open map – Map that satisfies a condition similar to that of being an open map.
Base (topology) – Collection of open sets used to define a topology
Clopen set – Subset which is both open and closed
Closed set – Complement of an open subset
Domain (mathematical analysis) – Connected open subset of a topological space
Local homeomorphism – Mathematical function revertible near each point
Open map – A function that sends open (resp. closed) subsets to open (resp. closed) subsetsPages displaying short descriptions of redirect targets
Subbase – Collection of subsets that generate a topology

Notes

^One exception if the if $X {\displaystyle X}$ is endowed with thediscrete topology, in which case every subset of $X {\displaystyle X}$ is both a regular open subset and a regular closed subset of $X . {\displaystyle X.}$

References

^Munkres 2000, pp. 76–77.
^Ueno, Kenji; et al. (2005)."The birth of manifolds".A Mathematical Gift: The Interplay Between Topology, Functions, Geometry, and Algebra. Vol. 3. American Mathematical Society. p. 38.ISBN 9780821832844.
^Munkres 2000, pp. 76.
^^a ^bTaylor, Joseph L. (2011)."Analytic functions".Complex Variables. The Sally Series. American Mathematical Society. p. 29.ISBN 9780821869017.
^Krantz, Steven G. (2009)."Fundamentals".Essentials of Topology With Applications. CRC Press. pp. 3–4.ISBN 9781420089745.
^Munkres 2000, pp. 88.
^Munkres 2000, pp. 95.
^Munkres 2000, pp. 102.
^^a ^b ^c ^d ^eHart 2004, p. 9.
^^a ^b ^c ^d ^e ^f ^g ^h ⁱHart 2004, pp. 8–9.
^Oxtoby, John C. (1980), "4. The Property of Baire",Measure and Category, Graduate Texts in Mathematics, vol. 2 (2nd ed.), Springer-Verlag, pp. 19–21,ISBN 978-0-387-90508-2.
^Kuratowski, Kazimierz (1966),Topology. Vol. 1, Academic Press and Polish Scientific Publishers.
^^a ^b ^c ^d ^e ^fHart 2004, p. 8.

Bibliography

Hart, Klaas (2004).Encyclopedia of general topology. Amsterdam Boston: Elsevier/North-Holland.ISBN 0-444-50355-2.OCLC 162131277.
Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2004).Encyclopedia of general topology. Elsevier.ISBN 978-0-444-50355-8.
Munkres, James R. (2000).Topology (2nd ed.).Upper Saddle River, NJ:Prentice Hall, Inc.ISBN 978-0-13-181629-9.OCLC 42683260.

External links

"Open set",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

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