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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 is defined—open sets are the sets that, with every point P, contain all points that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P).

- Motivation
- Definitions
- Euclidean space
- Metric space
- Topological space
- Special types of open sets
- Clopen sets and non-open and/or non-closed sets
- Regular open sets
- Properties
- Uses
- Notes and cautions
- "Open" is defined relative to a particular topology
- Generalizations of open sets
- See also
- Notes
- References
- Bibliography
- External links

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 but 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 measure 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: *d*(*x*, *y*) = |*x* − *y*|. 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 of the Euclidean *n*-space **R**^{n} is *open* if, for every point x in , there exists a positive real number ε (depending on x) such that a point in **R**^{n} belongs to as soon as its Euclidean distance from x is smaller than ε.^{ [1] } Equivalently, a subset of **R**^{n} is open if every point in is the center of an open ball contained in

A subset *U* of a metric space (*M*, *d*) is called *open* if, given any point *x* in *U*, there exists a real number *ε* > 0 such that, given any point satisfying *d*(*x*, *y*) < *ε*, *y* also belongs to *U*. Equivalently, *U* is open if every point in *U* has a neighborhood contained in *U*.

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 be a set. A family of subsets of is a *topology* on , and the elements of are the *open sets* of the topology if

- and (both and are open sets)
- then (any union of open sets is an open set)
- then (any finite intersection of open sets is an open set)

Infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form where is a positive integer, is the set 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.

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 *and* a closed subset. Such subsets are known as ** clopen sets **. Explicitly, a subset of a topological space is called

In *any* topological space the empty set and the set itself are always open. These two sets are the most well-known examples of clopen subsets and they show that clopen subsets exist in *every* topological space. To see why is clopen, begin by recalling that the sets and are, by definition, always open subsets (of ). Also by definition, a subset is called *closed* if (and only if) its complement in which is the set is an open subset. Because the complement (in ) of the entire set is the empty set (i.e. ), which is an open subset, this means that is a closed subset of (by definition of "closed subset"). Hence, no matter what topology is placed on the entire space is simultaneously both an open subset and also a closed subset of ; said differently, is *always* a clopen subset of Because the empty set's complement is which is an open subset, the same reasoning can be used to conclude that is also a clopen subset of

Consider the real line endowed with its usual Euclidean topology, whose open sets are defined as follows: every interval of real numbers belongs to the topology, every union of such intervals, e.g. belongs to the topology, and as always, both and belong to the topology.

- The interval is open in because it belongs to the Euclidean topology. If were to have an open complement, it would mean by definition that were closed. But does not have an open complement; its complement is which does
*not*belong to the Euclidean topology since it is not a union of open intervals of the form Hence, is an example of a set that is open but not closed. - By a similar argument, the interval is a closed subset but not an open subset.
- Finally, since neither nor its complement belongs to the Euclidean topology (because it can not be written as a union of intervals of the form ), this means that is neither open nor closed.

If a topological space is endowed with the discrete topology (so that by definition, every subset of is open) then every subset of is a clopen subset. For a more advanced example reminiscent of the discrete topology, suppose that is an ultrafilter on a non-empty set Then the union is a topology on with the property that *every* non-empty proper subset of is *either* an open subset or else a closed subset, but never both; that is, if (where ) then *exactly one* of the following two statements is true: either (1) or else, (2) Said differently, *every* subset is open or closed but the *only* subsets that are both (i.e. that are clopen) are and

A subset of a topological space is called a ** regular open set ** if or equivalently, if where (resp. ) denotes the topological boundary (resp. interior, closure) of in A topological space for which there exists a base consisting of regular open sets is called a

The union of any number of open sets, or infinitely many open sets, is open.^{ [2] } The intersection of a finite number of open sets is open.^{ [2] }

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.^{ [3] }

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*.

A function between two topological spaces and is * continuous * if the preimage of every open set in is open in The function is called * open * if the image of every open set in is open in

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 as "the topological space *X*" rather than "the topological space ", despite the fact that all the topological data is contained in If there are two topologies on the same set, a set *U* that is open in the first topology might fail to be open in the second topology. For example, if *X* is any topological space and *Y* is any subset of *X*, the set *Y* can be given its own topology (called the 'subspace topology') defined by "a set *U* is open in the subspace topology on *Y* if and only if *U* is the intersection of *Y* with an open set from the original topology on *X*." This potentially introduces new open sets: if *V* is open in the original topology on *X*, but isn't open in the original topology on *X*, then is open in the subspace topology on *Y*.

As a concrete example of this, if *U* is defined as the set of rational numbers in the interval then *U* is an open subset of the rational numbers, but not of the real numbers. This is because when the surrounding space is the rational numbers, for every point *x* in *U*, there exists a positive number *a* such that all *rational* points within distance *a* of *x* are also in *U*. On the other hand, when the surrounding space is the reals, then for every point *x* in *U* there is *no* positive *a* such that all *real* points within distance *a* of *x* are in *U* (because *U* contains no non-rational numbers).

Throughout, will be a topological space.

A subset of a topological space is called:

if , and the complement of such a set is called*α-open**α-closed*.^{ [4] },*preopen*, or*nearly open*if it satisfies any of the following equivalent conditions:*locally dense*^{ [5] }- There exists subsets such that is open in is a dense subset of and
^{ [5] } - There exists an open (in ) subset such that is a dense subset of
^{ [5] }

The complement of a preopen set is called

.*preclosed*if . The complement of a b-open set is called*b-open**b-closed*.^{ [4] }or*β-open*if it satisfies any of the following equivalent conditions:*semi-preopen*^{ [4] }- is a regular closed subset of
^{ [5] } - There exists a preopen subset of such that
^{ [5] }

The complement of a β-open set is called

.*β-closed*if it satisfies any of the following equivalent conditions:*sequentially open*- Whenever a sequence in converges to some point of then that sequence is eventually in Explicitly, this means that if is a sequence in and if there exists some is such that in then is eventually in (that is, there exists some integer such that if then ).
- is equal to its
in which by definition is the set*sequential interior*

The complement of a sequentially open set is called

. A subset is sequentially closed in if and only if is equal to its*sequentially closed*, which by definition is the set consisting of all for which there exists a sequence in that converges to (in ).*sequential closure*and is said to have*almost open*if there exists an open subset such that is a meager subset, where denotes the symmetric difference.*the Baire property*^{ [6] }- The subset is said to have
**the Baire property in the restricted sense**if for every subset of the intersection has the Baire property relative to .^{ [7] }

- The subset is said to have
if . The complement in of a semi-open set is called a*semi-open**semi-closed*set.^{ [8] }- The
(in ) of a subset denoted by is the intersection of all semi-closed subsets of that contain as a subset.*semi-closure*^{ [8] }

- The
if for each there exists some semiopen subset of such that*semi-θ-open*^{ [8] }(resp.*θ-open*) if its complement in is a θ-closed (resp.*δ-open**δ-closed*) set, where by definition, a subset of is called(resp.*θ-closed*) if it is equal to the set of all of its θ-cluster points (resp. δ-cluster points). A point is called a*δ-closed*(resp. a*θ-cluster point*) of a subset if for every open neighborhood of in the intersection is not empty (resp. is not empty).*δ-cluster point*^{ [8] }

Using the fact that

- and

whenever two subsets satisfy 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.^{ [5] } 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.^{ [5] } Preopen sets need not be semi-open and semi-open sets need not be preopen.^{ [5] }

Arbitrary unions of preopen (resp. α-open, b-open, semi-preopen) sets are once again preopen (resp. α-open, b-open, semi-preopen).^{ [9] } However, finite intersections of preopen sets need not be preopen.^{ [8] } The set of all α-open subsets of a space forms a topology on that is finer than ^{ [4] }

A topological space is Hausdorff if and only if every compact subspace of is θ-closed.^{ [8] } A space is totally 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 the ** closure ** of every preopen subset is open.

- Almost open map – A map that satisfies a condition similar to that of being an open map.
- Base (topology) – Collection of open sets that is sufficient for defining a topology
- Clopen set – Subset that is both open and closed
- Closed set – The complement of an open subset a topological space. It contains all points that are "close" to it.
- Local homeomorphism – Continuous open map that, around every point in its domain, has a neighborhood on which it restricts to a homomorphism
- Open map
- Subbase – Collection of subsets whose closure by finite intersections form the base of a topology

- ↑ One exception if the if is endowed with the discrete topology, in which case every subset of is both a regular open subset and a regular closed subset of

In mathematics, a **continuous function** is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be *discontinuous*. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.

In mathematics, a **topological space** is, roughly speaking, a geometrical space in which *closeness* is defined but, generally, cannot be measured by a numeric distance. More specifically, a topological space is a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In mathematics, the **closure** of a subset *S* of points in a topological space consists of all points in *S* together with all limit points of *S*. The closure of *S* may equivalently be defined as the union of *S* and its boundary, and also as the intersection of all closed sets containing *S*. Intuitively, the closure can be thought of as all the points that are either in *S* or "near" *S*. A point which is in the closure of *S* is a point of closure of *S*. The notion of closure is in many ways dual to the notion of interior.

In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

In geometry, topology, and related branches of mathematics, a **closed set** is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold.

In topology, a **discrete space** is a particularly simple example of a topological space or similar structure, one in which the points form a *discontinuous sequence*, meaning they are *isolated* from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology.

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

In mathematics, more specifically in topology, an **open map** is a function between two topological spaces that maps open sets to open sets. That is, a function is open if for any open set in the image is open in Likewise, a **closed map** is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.

In functional and convex analysis, and related disciplines of mathematics, the **polar set** is a special convex set associated to any subset of a vector space lying in the dual space The **bipolar** of a subset is the polar of but lies in .

In general topology and related areas of mathematics, the **final topology** on a set with respect to a family of functions from topological spaces into is the finest topology on that makes all those functions continuous.

A **semiregular space** is a topological space whose regular open sets form a base.

In mathematics, a topological space is usually defined in terms of open sets. However, there are many equivalent **characterizations of the category of topological spaces**. Each of these definitions provides a new way of thinking about topological concepts, and many of these have led to further lines of inquiry and generalisation.

In topology and related fields of mathematics, a **sequential space** is a topological space that satisfies a very weak axiom of countability.

In the field of topology, a **Fréchet–Urysohn space** is a topological space with the property that for every subset the closure of in is identical to the *sequential* closure of in Fréchet–Urysohn spaces are a special type of sequential space.

In topology, a subfield of mathematics, *filters* are special families of subsets of a set that can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called *ultrafilters* have many useful technical properties and they may often be used in place of arbitrary filters.

In functional analysis and related areas of mathematics, a **complete topological vector space** is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer to. The notion of "points that get progressively closer" is made rigorous by *Cauchy nets* or *Cauchy filters*, which are generalizations of *Cauchy sequences*, while "point towards which they all get closer to" means that this net or filter converges to Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for *all* TVSs, including those that are not metrizable or Hausdorff.

In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces. One method is used if the disk is bounded: in this case, the **auxiliary normed space** is with norm The other method is used if the disk is absorbing: in this case, the auxiliary normed space is the quotient space If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic.

In mathematics, a **convergence space**, also called a **generalized convergence**, is a set together with a relation called a *convergence* that satisfies certain properties relating elements of *X* with the family of filters on *X*. Convergence spaces generalize the notions of convergence that are found in point-set topology, including metric convergence and uniform convergence. Every topological space gives rise to a canonical convergence but there are convergences, known as *non-topological convergences*, that do not arise from any topological space. Examples of convergences that are in general non-topological include convergence in measure and almost everywhere convergence. Many topological properties have generalizations to convergence spaces.

In mathematics, specifically topology, a **sequence covering map** is any of a class of maps between topological spaces whose definitions all somehow relate sequences in the codomain with sequences in the domain. Examples include *sequentially quotient* maps, *sequence coverings*, *1-sequence coverings*, and *2-sequence coverings*. These classes of maps are closely related to sequential spaces. If the domain and/or codomain have certain additional topological properties then these definitions become equivalent to other well-known classes of maps, such as open maps or quotient maps, for example. In these situations, characterizations of such properties in terms of convergent sequences might provide benefits similar to those provided by, say for instance, the characterization of continuity in terms of sequential continuity or the characterization of compactness in terms of sequential compactness.

- ↑ Ueno, Kenji; et al. (2005). "The birth of manifolds".
*A Mathematical Gift: The Interplay Between Topology, Functions, Geometry, and Algebra*.**3**. American Mathematical Society. p. 38. ISBN 9780821832844. - 1 2 Taylor, 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. - 1 2 3 4 5 Hart 2004, p. 9.
- 1 2 3 4 5 6 7 8 Hart 2004, pp. 8–9.
- ↑ Oxtoby, John C. (1980), "4. The Property of Baire",
*Measure and Category*, Graduate Texts in Mathematics,**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. - 1 2 3 4 5 6 Hart 2004, p. 8.
- ↑ Hart 2004, pp. 8-9.

- 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.

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

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