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.

For a subset of a Euclidean space, is a point of closure of if every open ball centered at contains a point of (this point may be itself).

This definition generalizes to any subset of a metric space Fully expressed, for a metric space with metric is a point of closure of if for every there exists some such that the distance (again, is allowed). Another way to express this is to say that is a point of closure of if the distance

This definition generalizes to topological spaces by replacing "open ball" or "ball" with "neighbourhood". Let be a subset of a topological space Then is a * point of closure * or

The definition of a point of closure is closely related to the definition of a limit point. The difference between the two definitions is subtle but important – namely, in the definition of limit point, every neighbourhood of the point in question must contain a point of the set *other than itself*. The set of all limit points of a set is called the ** derived set ** of

Thus, every limit point is a point of closure, but not every point of closure is a limit point. A point of closure which is not a limit point is an isolated point. In other words, a point is an isolated point of if it is an element of and if there is a neighbourhood of which contains no other points of other than itself.^{ [2] }

For a given set and point is a point of closure of if and only if is an element of or is a limit point of (or both).

The * closure* of a subset of a topological space denoted by or possibly by (if is understood), where if both and are clear from context then it may also be denoted by or (moreover, is sometimes capitalized to ) can be defined using any of the following equivalent definitions:

- is the set of all points of closure of
- is the set together with all of its limit points.
^{ [3] } - is the intersection of all closed sets containing
- is the smallest closed set containing
- is the union of and its boundary
- is the set of all for which there exists a net (valued) in that converges to in

The closure of a set has the following properties.^{ [4] }

- is a closed superset of
- The set is closed if and only if
- If then is a subset of
- If is a closed set, then contains if and only if contains

Sometimes the second or third property above is taken as the *definition* of the topological closure, which still make sense when applied to other types of closures (see below).^{ [5] }

In a first-countable space (such as a metric space), is the set of all limits of all convergent sequences of points in For a general topological space, this statement remains true if one replaces "sequence" by "net" or "filter".

Note that these properties are also satisfied if "closure", "superset", "intersection", "contains/containing", "smallest" and "closed" are replaced by "interior", "subset", "union", "contained in", "largest", and "open". For more on this matter, see closure operator below.

Consider a sphere in 3 dimensions. Implicitly there are two regions of interest created by this sphere; the sphere itself and its interior (which is called an open 3-ball). It is useful to be able to distinguish between the interior of 3-ball and the surface, so we distinguish between the open 3-ball, and the closed 3-ball – the closure of the 3-ball. The closure of the open 3-ball is the open 3-ball plus the surface.

- In any space,
- In any space

Giving and the standard (metric) topology:

- If is the Euclidean space of real numbers, then
- If is the Euclidean space then the closure of the set of rational numbers is the whole space We say that is dense in
- If is the complex plane then
- If is a finite subset of a Euclidean space then (For a general topological space, this property is equivalent to the T
_{1}axiom.)

On the set of real numbers one can put other topologies rather than the standard one.

- If is endowed with the lower limit topology, then
- If one considers on the discrete topology in which every set is closed (open), then
- If one considers on the trivial topology in which the only closed (open) sets are the empty set and itself, then

These examples show that the closure of a set depends upon the topology of the underlying space. The last two examples are special cases of the following.

- In any discrete space, since every set is closed (and also open), every set is equal to its closure.
- In any indiscrete space since the only closed sets are the empty set and itself, we have that the closure of the empty set is the empty set, and for every non-empty subset of In other words, every non-empty subset of an indiscrete space is dense.

The closure of a set also depends upon in which space we are taking the closure. For example, if is the set of rational numbers, with the usual relative topology induced by the Euclidean space and if then is both closed and open in because neither nor its complement can contain , which would be the lower bound of , but cannot be in because is irrational. So, has no well defined closure due to boundary elements not being in . However, if we instead define to be the set of real numbers and define the interval in the same way then the closure of that interval is well defined and would be the set of all *real numbers* greater than *or equal to*.

A * closure operator* on a set is a mapping of the power set of , into itself which satisfies the Kuratowski closure axioms. Given a topological space , the topological closure induces a function that is defined by sending a subset to where the notation or may be used instead. Conversely, if is a closure operator on a set then a topological space is obtained by defining the closed sets as being exactly those subsets that satisfy (so complements in of these subsets form the open sets of the topology).

The closure operator is dual to the interior operator, which is denoted by in the sense that

and also

Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be readily translated into the language of interior operators by replacing sets with their complements in

In general, the closure operator does not commute with intersections. However, in a complete metric space the following result does hold:

**Theorem ^{ [7] }** (C. Ursescu) — Let be a sequence of subsets of a complete metric space

- If each is closed in then
- If each is open in then

A subset is closed in if and only if In particular:

- The closure of the empty set is the empty set;
- The closure of itself is
- The closure of an intersection of sets is always a subset of (but need not be equal to) the intersection of the closures of the sets.
- In a union of finitely many sets, the closure of the union and the union of the closures are equal; the union of zero sets is the empty set, and so this statement contains the earlier statement about the closure of the empty set as a special case.
- The closure of the union of infinitely many sets need not equal the union of the closures, but it is always a superset of the union of the closures.

If and if is a subspace of (meaning that is endowed with the subspace topology that induces on it), then and the closure of computed in is equal to the intersection of and the closure of computed in :

^{ [proof 1] }

In particular, is dense in if and only if is a subset of

If but is not necessarily a subset of then only

is guaranteed in general, where this containment could be strict (consider for instance with the usual topology, and ^{ [proof 2] }) although if is an open subset of then the equality will hold^{ [proof 3] } (no matter the relationship between and ). Consequently, if is any open cover of and if is any subset then:

because for every (where every is endowed with the subspace topology induced on it by ). This equality is particularly useful when is a manifold and the sets in the open cover are domains of coordinate charts. In words, this result shows that the closure in of any subset can be computed "locally" in the sets of any open cover of and then unioned together. In this way, this result can be viewed as the analogue of the well-known fact that a subset is closed in if and only if it is "locally closed in ", meaning that if is any open cover of then is closed in if and only if is closed in for every

One may elegantly define the closure operator in terms of universal arrows, as follows.

The powerset of a set may be realized as a partial order category in which the objects are subsets and the morphisms are inclusion maps whenever is a subset of Furthermore, a topology on is a subcategory of with inclusion functor The set of closed subsets containing a fixed subset can be identified with the comma category This category — also a partial order — then has initial object Thus there is a universal arrow from to given by the inclusion

Similarly, since every closed set containing corresponds with an open set contained in we can interpret the category as the set of open subsets contained in with terminal object the interior of

All properties of the closure can be derived from this definition and a few properties of the above categories. Moreover, this definition makes precise the analogy between the topological closure and other types of closures (for example algebraic closure), since all are examples of universal arrows.

- Adherent point – An point that belongs to the closure of some give subset of a topological space.
- Closure algebra
- Derived set (mathematics)
- Interior (topology)
- Limit point – A point
*x*in a topological space, all of whose neighborhoods contain some other point in a given subset that is different from*x*

- ↑ Because is a closed subset of the intersection is a closed subset of (by definition of the subspace topology), which implies that (because is the
*smallest*closed subset of containing ). Because is a closed subset of from the definition of the subspace topology, there must exist some set such that is closed in and Because and is closed in the minimality of implies that Intersecting both sides with shows that - ↑ From and it follows that and which implies
- ↑ Let and assume that is open in Let which is equal to (because ). The complement is open in where being open in now implies that is also open in Consequently is a closed subset of where contains as a subset (because if is in then ), which implies that Intersecting both sides with proves that The reverse inclusion follows from

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, **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.

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 mathematics, a subset of a topological space is called **nowhere dense** or **rare** if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered anywhere. For example, the integers are nowhere dense among the reals, whereas an open ball is not.

In mathematics, specifically in topology, the **interior** of a subset S of a topological space X is the union of all subsets of S that are open in X. A point that is in the interior of S is an **interior point** of S.

In topology and mathematics in general, the **boundary** of a subset *S* of a topological space *X* is the set of points which can be approached both from *S* and from the outside of *S*. More precisely, it is the set of points in the closure of not belonging to the interior of An element of the boundary of is called a **boundary point** of The term **boundary operation** refers to finding or taking the boundary of a set. Notations used for boundary of a set include and Some authors use the term **frontier** instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, *Metric Spaces* by E. T. Copson uses the term boundary to refer to Hausdorff's **border**, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term **residue**, which is defined as the intersection of a set with the closure of the border of its complement.

In mathematics, a **limit point** of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with respect to the topology on also contains a point of other than itself. A limit point of a set does not itself have to be an element of There is also a closely related concept for sequences. A **cluster point** or **accumulation point** of a sequence in a topological space is a point such that, for every neighbourhood of there are infinitely many natural numbers such that This definition of a cluster or accumulation point of a sequence generalizes to nets and filters. In contrast to sets, for a sequence, net, or filter, the term "limit point" is *not* synonymous with a "cluster/accumulation point". The similarly named notions of a limit point of a filter, a limit point of a sequence, or a limit point of a net; each of these respectively refers to a point that a filter, sequence, or net converges to.

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 topology, the **exterior** of a subset of a topological space is the union of all open sets of which are disjoint from It is itself an open set and is disjoint from The exterior of in is often denoted by or, if is clear from context, then possibly also by or

In mathematics, more specifically in point-set topology, the **derived set** of a subset of a topological space is the set of all limit points of It is usually denoted by

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 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 mathematical field of topology, a **hyperconnected space** or **irreducible space** is a topological space *X* that cannot be written as the union of two proper closed sets. The name *irreducible space* is preferred in algebraic geometry.

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.

- ↑ Schubert 1968 , p. 20
- ↑ Kuratowski 1966 , p. 75
- ↑ Hocking & Young 1988 , p. 4
- ↑ Croom 1989 , p. 104
- ↑ Gemignani 1990 , p. 55, Pervin 1965 , p. 40 and Baker 1991 , p. 38 use the second property as the definition.
- ↑ Pervin 1965 , p. 41
- ↑ Zălinescu 2002, p. 33.

- Baker, Crump W. (1991),
*Introduction to Topology*, Wm. C. Brown Publisher, ISBN 0-697-05972-3 - Croom, Fred H. (1989),
*Principles of Topology*, Saunders College Publishing, ISBN 0-03-012813-7 - Gemignani, Michael C. (1990) [1967],
*Elementary Topology*(2nd ed.), Dover, ISBN 0-486-66522-4 - Hocking, John G.; Young, Gail S. (1988) [1961],
*Topology*, Dover, ISBN 0-486-65676-4 - Kuratowski, K. (1966),
*Topology*,**I**, Academic Press - Pervin, William J. (1965),
*Foundations of General Topology*, Academic Press - Schubert, Horst (1968),
*Topology*, Allyn and Bacon - Zălinescu, Constantin (30 July 2002).
*Convex Analysis in General Vector Spaces*. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112.

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

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