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In mathematics, **open sets** are a generalization of open intervals in the real line. In a metric space—that is, when a distance function is defined—open sets are the sets that, with every point, contain all points that are sufficiently near to (that is, all points whose distance to is less than some value depending on).

More generally, one defines open sets as the members of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, *every* subset can be open (the discrete topology), or no set can be open except the space itself and the empty set (the indiscrete topology).

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 as continuity, connectedness, and compactness, which were originally defined by means of a distance.

The most common case of a topology without any distance is given by manifolds, which are topological spaces that, *near* each point, resemble an open set of a Euclidean space, but on which no distance is defined in general. Less intuitive topologies are used in other branches of mathematics; for example, the Zariski topology, which is fundamental in algebraic geometry and scheme theory.

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

In the set of all real numbers, one has the natural Euclidean metric; that is, a function which measures the distance between two real numbers: . Therefore, given a real number *x*, one can speak of the set of all points close to that real number; that is, within *ε* of *x*. In essence, points within ε of *x* approximate *x* to an accuracy of degree *ε*. Note that *ε* > 0 always but as *ε* becomes smaller and smaller, one obtains points that approximate *x* to a higher and higher degree of accuracy. For example, if *x* = 0 and *ε* = 1, the points within *ε* of *x* are precisely the points of the interval (−1, 1); that is, the set of all real numbers between −1 and 1. However, with *ε* = 0.5, the points within *ε* of *x* are precisely the points of (−0.5, 0.5). Clearly, these points approximate *x* to a greater degree of accuracy than when *ε* = 1.

The previous discussion shows, for the case *x* = 0, that one may approximate *x* to higher and higher degrees of accuracy by defining *ε* to be smaller and smaller. In particular, sets of the form (−*ε*, *ε*) give us a lot of information about points close to *x* = 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to *x*. 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 define **R** 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 of **R**. 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 in **R** are equally close to 0, while any item that is not in **R** is not close to 0.

In general, one refers to the family of sets containing 0, used to approximate 0, as a ** neighborhood basis**; a member of this neighborhood basis is referred to as an

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

A subset

*U*

*U*

*U*

*U*

*U*

*U.*

A subset *U* of a metric space is called *open* if, given any point *x* in *U*, there exists a real number *ε* > 0 such that, given any point

*y**\in**M*

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

A topological space is a set on which a topology is defined, which consists of a collection of subsets that are said to be *open*, and satisfy the axioms given below.

More precisely, let

*X*

*\tau*

*X*

*X*

*\tau*

*X**\in**\tau*

*\varnothing**\in**\tau*

*X*

*\varnothing*

*\left\{**U*_{i}*:**i**\in**I**\right\}**\subseteq**\tau*

c*up*_{i}*U*_{i}*\in**\tau*

*U*_{1,}*\ldots,**U*_{n}*\in**\tau*

*U*_{1}*\cap* … *\cap**U*_{n}*\in**\tau*

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),*

*n*

*\{*0*\}*

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.

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 subset a closed subset. Such subsets are known as . Explicitly, a subset

*S*

*(X,**\tau)*

*S*

*X**\setminus**S*

*(X,**\tau)*

*S**\in**\tau*

*X**\setminus**S**\in**\tau.*

In topological space

*(X,**\tau),*

*\varnothing*

*X*

*X*

*X*

*\varnothing*

*X*

*S*

*X,*

*X**\setminus**S,*

*X*

*S**:*=*X*

*X**\setminus**S*=*\varnothing*

*S*=*X*

*X*

*X,*

*X*

*X*

*X*

*X.*

*X**\setminus**\varnothing*=*X,*

*S**:*=*\varnothing*

*X.*

Consider the real line

*\R*

*(a,**b)*

*(a,**b)**\cup**(c,**d),*

*\R*

*\varnothing*

- The interval

*I*=*(*0*,*1*)*

*\R*

*I*

*I*

*I*

*\R**\setminus**I*=*(*-inf*ty,*0*]**\cup**[*1*,*inf*ty),*

*(a,**b).*

*I*

- By a similar argument, the interval

*J*=*[*0*,*1*]*

- Finally, since neither

*K*=*[*0*,*1*)*

*\R**\setminus**K*=*(*-inf*ty,*0*)**\cup**[*1*,*inf*ty)*

*(a,**b)*

*K*

If a topological space

*X*

*X*

*X*

l{U}

*X.*

*\tau**:*=l{U}*\cup**\{**\varnothing**\}*

*X*

*S*

*X*

*\varnothing* ≠ *S**\subsetneq**X*

*S* ≠ *X*

*S**\in**\tau*

*X**\setminus**S**\in**\tau.*

*\varnothing*

*X.*

A subset

*S*

*X*

*\operatorname{Int}**\left(**\overline{S}**\right)*=*S*

*\operatorname{Bd}**\left(**\overline{S}**\right)*=*\operatorname{Bd}**S,*

*\operatorname{Bd}**S*

*\operatorname{Int}**S,*

*\overline{S}*

*S*

*X.*

*X*

*X*

*S*

*X*

*\overline{\operatorname{Int}**S}*=*S*

*\operatorname{Bd}**\left(**\operatorname{Int}**S**\right)*=*\operatorname{Bd}**S.*

The union of any number of open sets, or infinitely many open sets, is open.^{[3]} The intersection of a finite number of open sets is open.

A complement of an open set (relative to the space that the topology is defined on) is called a closed set. A set may be both open and closed (a clopen set). The empty set and the full space are examples of sets that are both open and closed.^{[4]}

Open sets have a fundamental importance in topology. The concept is required to define and make sense of topological space and other topological structures that deal with the notions of closeness and convergence for spaces such as metric spaces and uniform spaces.

Every subset *A* of a topological space *X* contains a (possibly empty) open set; the maximum (ordered under inclusion) such open set is called the interior of *A*. It can be constructed by taking the union of all the open sets contained in *A*.

*f**:**X**\to**Y*

*X*

*Y*

*Y*

*X.*

*f**:**X**\to**Y*

*X*

*Y.*

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

Whether a set is open depends on the topology under consideration. Having opted for greater brevity over greater clarity, we refer to a set *X* endowed with a topology

*\tau*

*(X,**\tau)*

*\tau.*

*V**\cap**Y*

*V**\cap**Y*

As a concrete example of this, if *U* is defined as the set of rational numbers in the interval

*(*0*,*1*),*

See also: Almost open map and Glossary of topology.

Throughout,

*(X,**\tau)*

A subset

*A**\subseteq**X*

*X*

- if
, and the complement of such a set is called
*A**~\subseteq~**\operatorname{int}*_{X}*\left(**\operatorname{cl}*_{X}*\left(**\operatorname{int}*_{X}*A**\right)**\right)***.** **,****, or****if it satisfies any of the following equivalent conditions:***A**~\subseteq~**\operatorname{int}*_{X}*\left(**\operatorname{cl}*_{X}*A**\right).*- There exists subsets such that
*D,**U**\subseteq**X*is open in*U**X,*is a dense subset of*D*and*X,**A*=*U**\cap**D.* - There exists an open (in ) subset
*X*such that*U**\subseteq**X*is a dense subset of*A**U.*

**.****if**.. The complement of a b-open set is called*A**~\subseteq~**\operatorname{int}*_{X}*\left(**\operatorname{cl}*_{X}*A**\right)**~\cup~**\operatorname{cl}*_{X}*\left(**\operatorname{int}*_{X}*A**\right)*- or if it satisfies any of the following equivalent conditions:
*A**~\subseteq~**\operatorname{cl}*_{X}*\left(**\operatorname{int}*_{X}*\left(**\operatorname{cl}*_{X}*A**\right)**\right)*- is a regular closed subset of
*\operatorname{cl}*_{X}*A**X.* - There exists a preopen subset of
*U*such that*X**U**\subseteq**A**\subseteq**\operatorname{cl}*_{X}*U.*

- if it satisfies any of the following equivalent conditions:
- Whenever a sequence in converges to some point of
*X*then that sequence is eventually in*A,*Explicitly, this means that if*A.*is a sequence in*x*_{\bull}=*\left(**x*_{i}inf *ty**\right)**i*=1and if there exists some*X*is such that*a**\in**A*in*x*_{\bull}*\to**x*then*(X,**\tau),*is eventually in*x*_{\bull}(that is, there exists some integer*A*such that if*i*then*j**\geq**i,*).*x*_{j}*\in**A* - is equal to its in
*A*which by definition is the set*X,**\begin{alignat}{*4*} \operatorname{SeqInt}*_{X}*A :&*=*\{**a**\in**A**~:~*wheneverasequencein*X*convergesto*a*in*(X,**\tau),*thenthatsequenceiseventuallyin*A**\}**\\ &*=*\{**a**\in**A**~:~*theredoesNOTexistasequencein*X**\setminus**A*thatconvergesin*(X,**\tau)*toapointin*A**\}**\\ \end{alignat}*

is sequentially closed in*S**\subseteq**X*if and only if*X*is equal to its , which by definition is the set*S*consisting of all*\operatorname{SeqCl}*_{X}*S*for which there exists a sequence in*x**\in**X*that converges to*S*(in*x*).*X* - Whenever a sequence in
- and is said to have if there exists an open subset such that
*U**\subseteq**X*is a meager subset, where*A*t*riangleup**U*denotes the symmetric difference.t

*riangleup*^{[5]}- The subset

is said to have*A**\subseteq**X***the Baire property in the restricted sense**if for every subsetof*E*the intersection*X*has the Baire property relative to*A\cap**E*.*E*^{[6]} - if . The complement in
*A**~\subseteq~**\operatorname{cl}*_{X}*\left(**\operatorname{int}*_{X}*A**\right)*of a semi-open set is called a*X***set.**- The (in

) of a subset*X*denoted by*A**\subseteq**X,*is the intersection of all semi-closed subsets of*\operatorname{sCl}*_{X}*A,*that contain*X*as a subset.*A* - if for each there exists some semiopen subset
*x**\in**A*of*U*such that*X**x**\in**U**\subseteq**\operatorname{sCl}*_{X}*U**\subseteq**A.* - (resp. ) if its complement in is a θ-closed (resp.) set, where by definition, a subset of
*X*is called (resp. ) if it is equal to the set of all of its θ-cluster points (resp. δ-cluster points). A point*X*is called a (resp. a ) of a subset*x**\in**X*if for every open neighborhood*B**\subseteq**X*of*U*in*x*the intersection*X,*is not empty (resp.*B**\cap**\operatorname{cl}*_{X}*U*is not empty).*B**\cap**\operatorname{int}*_{X\left(}*\operatorname{cl}*_{X}*U**\right)*

Using the fact that

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

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

whenever two subsets

*A,**B**\subseteq**X*

*A**\subseteq**B,*

- 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. 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. Preopen sets need not be semi-open and semi-open sets need not be preopen.

Arbitrary unions of preopen (resp. α-open, b-open, semi-preopen) sets are once again preopen (resp. α-open, b-open, semi-preopen). However, finite intersections of preopen sets need not be preopen. The set of all α-open subsets of a space

*(X,**\tau)*

*X*

*\tau.*

A topological space

*X*

*X*

*X*

- Book: Hart, Klaas . Encyclopedia of general topology . Elsevier/North-Holland . Amsterdam Boston . 2004 . 0-444-50355-2 . 162131277 .
- Book: Encyclopedia of general topology . Klaas Pieter . Hart . Jun-iti . Nagata . Jerry E. . Vaughan . Elsevier . 2004 . 978-0-444-50355-8 .

- Book: Ueno . Kenji . etal . 2005 . A Mathematical Gift: The Interplay Between Topology, Functions, Geometry, and Algebra . The birth of manifolds . 3 . American Mathematical Society . 9780821832844 . 38 . https://books.google.com/books?id=GCHwtdj8MdEC&pg=PA38.
- One exception if the if is endowed with the discrete topology, in which case every subset of
*X*is both a regular open subset and a regular closed subset of*X**X.* - Book: Taylor, Joseph L. . 2011 . Complex Variables . Analytic functions . The Sally Series . American Mathematical Society . 9780821869017 . 29 . https://books.google.com/books?id=NHcdl0a7Ao8C&pg=PA29.
- Book: Krantz, Steven G. . Steven G. Krantz . 2009 . Essentials of Topology With Applications . Fundamentals . CRC Press . 9781420089745 . 3–4 . https://books.google.com/books?id=LUhabKjfQZYC&pg=PA3.
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