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Encyclopedia > Closure (topology)

In mathematics, the closure of a set S consists of all points which are intuitively "close to 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 other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, duality has numerous meanings. ... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...

Contents

Definitions

Point of closure

For S a subset of an Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S. (This point may be x itself and x needn't be in S.) Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ...


This definition generalises to any subset S of a metric space X. Fully expressed, for X a metric space with metric d, x is a point of closure of S if for every r > 0, there is a y in S such that the distance d(x, y) < r. (Again, we may have x = y.) Another way to express this is to say that x is a point of closure of S if the distance d(x, S) := inf{d(x, s) : s in S} = 0. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all other elements of the subset. ...


This definition generalises to topological spaces by replacing "open ball" or "ball" with "neighbourhood". Let S be a subset of a topological space X. Then x is a point of closure of S if every neighbourhood of x contains a point of S. Note that this definition does not depend upon whether neighbourhoods are required to be open. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... This is a glossary of some terms used in the branch of mathematics known as topology. ...


Limit point

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 neighborhood of the point x in question must contain a point of the set other than x itself. In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ...


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 x is an isolated point of S if it is an element of S and if there is a neighbourhood of x which contains no other points of S other than x itself. In topology, a branch of mathematics, a point x of a set S is called an isolated point, if there exists a neighborhood of x not containing other points of S. In particular, in a Euclidean space (or in a metric space), x is an isolated point of S, if...


For a given set S and point x, x is a point of closure of S if and only if x is an element of S or x is a limit point of S. ↔ ⇔ ≡ logical symbols representing iff. ...


Closure of a set

The closure of a set S is the set of all points of closure of S. The closure of S is denoted cl(S), Cl(S), or bar{S}. The closure of a set has the following properties.

  • cl(S) is a closed superset of S.
  • cl(S) is the intersection of all closed sets containing S.
  • cl(S) is the smallest closed set containing S.
  • A set S is closed if and only if S = cl(S).
  • If S is a subset of T, then cl(S) is a subset of cl(T).
  • If A is a closed set, then A contains S if and only if A contains cl(S).

Sometimes the second or third property above is taken as the definition of the topological closure. In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... ↔ ⇔ ≡ logical symbols representing iff. ...


In a first-countable space (such as a metric space), cl(S) is the set of all limits of all convergent sequences of points in S. For a general topological space, this statement remains true if one replaces "sequence" by "net" or "filter". In topology, a first-countable space is a topological space satisfying the first axiom of countability. Specifically, a space X is said to be first-countable if each point has a countable local base. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... For other senses of this word, see sequence (disambiguation). ... In topology and related areas of mathematics a net or Moore-Smith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces. ... In mathematics, a filter is a special subset of a partially ordered set. ...


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. In mathematics, the closure of a set S consists of all points which are intuitively close to 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. ...


Examples

  • In any space, the closure of the empty set is the empty set.
  • In any space X, X = cl(X).
  • If X is the Euclidean space R of real numbers, then cl((0, 1)) = [0, 1].
  • If X is the Euclidean space R, then the closure of the set Q of rational numbers is the whole space R. We say that Q is dense in R.
  • If X is the complex plane C = R2, then cl({z in C : |z| > 1}) = {z in C : |z| ≥ 1}.
  • If S is a finite subset of a Euclidean space, then cl(S) = S. (For a general topological space, this property is equivalent to the T1 axiom.)

On the set of real numbers one can put other topologies rather than the standard one. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, the term dense has at least three different meanings. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... The title given to this article is incorrect due to technical limitations. ...

  • If X = R, where R has the lower limit topology, then cl((0, 1)) = [0, 1).
  • If one considers on R the topology in which every set is open (closed), then cl((0, 1)) = (0, 1).
  • If one considers on R the topology in which the only open (closed) sets are the empty set and R itself, then cl((0, 1)) = R.

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 mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties. ...

  • In any discrete space, since every set is open (closed), every set is equal to its closure.
  • In any indiscrete space X, since the only open (closed) sets are the empty set and X itself, we have that the closure of the empty set is the empty set, and for every non-empty subset A of X, cl(A) = X. 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 X is the set of rational numbers, with the usual subspace topology induced by the Euclidean space R, and if S = {q in Q : q2 > 2}, then S is closed in Q, and the closure of S in Q is S; however, the closure of S in the Euclidean space R is the set of all real numbers greater than or equal to sqrt2. In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. ... In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology). ...


Closure operator

The closure operator is dual to the interior operator o, in the sense that In mathematics, duality has numerous meanings. ... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...

S = X (X S)o

and also

So = X (X S)

where X denotes the topological space containing S, and the backslash refers to the set-theoretic difference. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...


Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be easily translated into the language of interior operators, by replacing sets with their complements. In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...


At most 14 distinct sets can be obtained by repeatedly applying the operations of closure and complement to a given set, a result known as the Kuratowski's closure-complement problem. At most 14 distinct sets can be obtained by repeatedly applying the set operations of closure and complement to a given seed set in a topological space. ...


Facts about closures

The set S is closed if and only if Cl(S) = S. In particular, the closure of the empty set is the empty set, and the closure of X itself is X. 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. In topology and related branches of mathematics, a closed set is a set whose complement is open. ... The empty set is the set containing no elements. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... Superset redirects here. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ...


If A is a subspace of X containing S, then the closure of S computed in A is equal to the intersection of A and the closure of S computed in X: Cl_A(S) = Acap Cl_X(S). In particular, S is dense in A iff A is a subset of ClX(S). Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...


See also

In abstract algebra, an interior algebra is an algebraic structure of the signature <A, ·, +, , 0, 1, I> where <A, ·, +, , 0, 1> is a Boolean algebra and I is a unary operator, the interior operator, satisfying the identities: xI ≤ x xII = xI (xy)I = xIyI 1I = 1 xI is called the...

  Results from FactBites:
 
Closure (topology) - Biocrawler (1170 words)
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.
Kids.Net.Au - Encyclopedia > Closure (topology) (312 words)
In topology and mathematical analysis, the closure of a subset
In a union of finitely many sets, the closure of the union and the union of the closures are equal; for infinitely many sets, this need not be the case.
However in any case, the closure of a union of sets is always a superset of the union of the closures of the sets.
  More results at FactBites »

 
 

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