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# Closure (mathematics)

A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation. 

62 relations: Abstract algebra, Algebra of sets, Algebraic closure, Arity, Axiom, Binary relation, Clopen set, Closed set, Closure (topology), Commutative algebra, Conjugate closure, Convex hull, Convex set, Countable set, Field (mathematics), First-countable space, Formal language, Galois connection, Generating set of a group, Geometry, Group (mathematics), Group theory, Idempotence, If and only if, Integer, Integral element, Interval (mathematics), Inverse element, Kleene star, Kuratowski closure axioms, Limit of a sequence, Limit point, Linear algebra, Linear combination, Linear span, Linear subspace, Mathematical analysis, Matroid, Monad (category theory), Net (mathematics), Open set, Operation (mathematics), Ordinal number, Predicate (mathematical logic), Preorder, Probability theory, Rewriting, Set (mathematics), Set theory, Sigma-algebra, ... Expand index (12 more) »

## Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

## Algebra of sets

The algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion.

## Algebraic closure

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

## Arity

In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes.

## Axiom

An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

## Binary relation

In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.

## Clopen set

In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed.

## Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

## Closure (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.

## Commutative algebra

Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings.

## Conjugate closure

In group theory, the conjugate closure of a subset S of a group G is the subgroup of G generated by SG, i.e. the closure of SG under the group operation, where SG is the set of the conjugates of the elements of S: The conjugate closure of S is denoted G> or G. The conjugate closure of any subset S of a group G is always a normal subgroup of G; in fact, it is the smallest (by inclusion) normal subgroup of G which contains S. For this reason, the conjugate closure is also called the normal closure of S or the normal subgroup generated by S. The normal closure can also be characterized as the intersection of all normal subgroups of G which contain S. Any normal subgroup is equal to its normal closure.

## Convex hull

In mathematics, the convex hull or convex envelope or convex closure of a set X of points in the Euclidean plane or in a Euclidean space (or, more generally, in an affine space over the reals) is the smallest convex set that contains X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around X., p. 3.

## Convex set

In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations.

## Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## First-countable space

In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability".

## Formal language

In mathematics, computer science, and linguistics, a formal language is a set of strings of symbols together with a set of rules that are specific to it.

## Galois connection

In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets).

## Generating set of a group

In abstract algebra, a generating set of a group is a subset such that every element of the group can be expressed as the combination (under the group operation) of finitely many elements of the subset and their inverses.

## Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

## Idempotence

Idempotence is the property of certain operations in mathematics and computer science that they can be applied multiple times without changing the result beyond the initial application.

## If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Integral element

In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and a_j \in A such that That is to say, b is a root of a monic polynomial over A. If every element of B is integral over A, then it is said that B is integral over A, or equivalently B is an integral extension of A. If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial).

## Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.

## Inverse element

In abstract algebra, the idea of an inverse element generalises concepts of a negation (sign reversal) in relation to addition, and a reciprocal in relation to multiplication.

## Kleene star

In mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters.

## Kuratowski closure axioms

In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set.

## Limit of a sequence

As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1.

## Limit point

In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself.

## Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

## Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

## Linear span

In linear algebra, the linear span (also called the linear hull or just span) of a set of vectors in a vector space is the intersection of all subspaces containing that set.

## Linear subspace

In linear algebra and related fields of mathematics, a linear subspace, also known as a vector subspace, or, in the older literature, a linear manifold, is a vector space that is a subset of some other (higher-dimension) vector space.

## Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

## Matroid

In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces.

In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is an endofunctor (a functor mapping a category to itself), together with two natural transformations.

## Net (mathematics)

In mathematics, more specifically in general topology and related branches, a net or Moore&ndash;Smith sequence is a generalization of the notion of a sequence.

## Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

## Operation (mathematics)

In mathematics, an operation is a calculation from zero or more input values (called operands) to an output value.

## Ordinal number

In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.

## Predicate (mathematical logic)

In mathematical logic, a predicate is commonly understood to be a Boolean-valued function P: X→, called the predicate on X. However, predicates have many different uses and interpretations in mathematics and logic, and their precise definition, meaning and use will vary from theory to theory.

## Preorder

In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive.

## Probability theory

Probability theory is the branch of mathematics concerned with probability.

## Rewriting

In mathematics, computer science, and logic, rewriting covers a wide range of (potentially non-deterministic) methods of replacing subterms of a formula with other terms.

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

## Set theory

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.

## Sigma-algebra

In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that includes the empty subset, is closed under complement, and is closed under countable unions and countable intersections.

## Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

## Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

## Symmetric closure

In mathematics, the symmetric closure of a binary relation R on a set X is the smallest symmetric relation on X that contains R. For example, if X is a set of airports and xRy means "there is a direct flight from airport x to airport y", then the symmetric closure of R is the relation "there is a direct flight either from x to y or from y to x".

## Symmetric relation

In mathematics and other areas, a binary relation R over a set X is symmetric if it holds for all a and b in X that a is related to b if and only if b is related to a. In mathematical notation, this is: Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.

## Term algebra

In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature.

## Tight closure

In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic.

## Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

## Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

## Transitive closure

In mathematics, the transitive closure of a binary relation R on a set X is the smallest relation on X that contains R and is transitive.

## Unary operation

In mathematics, a unary operation is an operation with only one operand, i.e. a single input.

## Upper set

In mathematics, an upper set (also called an upward closed set or just an upset) of a partially ordered set (X,≤) is a subset U with the property that, if x is in U and x≤y, then y is in U. The dual notion is lower set (alternatively, down set, decreasing set, initial segment, semi-ideal; the set is downward closed), which is a subset L with the property that, if x is in L and y≤x, then y is in L. The terms order ideal or ideal are sometimes used as synonyms for lower set.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

## References

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