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

In mathematics, a nowhere dense set in a topological space is a set whose closure has empty interior.

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 S, not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. Notations used for boundary of a set S include bd ( S ), fr ( S ), and ∂ S.

In mathematics, the transitive closure of a binary relation R on a set X is the transitive relation R < sup >+</ sup > on set X such that R < sup >+</ sup > contains R and R < sup >+</ sup > is minimal ( Lidl and Pilz 1998: 337 ).

For example, " every field has an algebraic closure " is not provable in ZF set theory, but the restricted form " every countable field has an algebraic closure " is provable in RCA < sub > 0 </ sub >, the weakest system typically employed in reverse mathematics.

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 mathematics, the support of a function is the set of points where the function is not zero-valued, or the closure of that set.

In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set.

In mathematics, a closure operator on a set S is a function from the power set of S to itself which satisfies the following conditions for all sets

In mathematics, a relatively compact subspace ( or relatively compact subset ) Y of a topological space X is a subset whose closure is compact.

The term normal closure is used in two senses in mathematics:

In mathematics, the upper topology on a partially ordered set X is the coarsest topology in which the closure of a singleton is the order section for each.

In mathematics, particularly measure theory, a σ-ideal of a sigma-algebra ( σ, read " sigma ," means countable in this context ) is a subset with certain desirable closure properties.

In mathematics, a Severi – Brauer variety over a field K is an algebraic variety V which becomes isomorphic to projective space over an algebraic closure of K. Examples are conic sections C: provided C is non-singular, it becomes isomorphic to the projective line over any extension field L over which C has a point defined.

# REDIRECT closure ( mathematics )

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

In mathematics, given a set and an equivalence relation on, the equivalence class of an element in is the subset of all elements in which are equivalent to.

* In mathematics, a certain kind of subset of a partially ordered set.

** Filter ( mathematics ), a special subset of a partially ordered set

* Interval ( mathematics ), a range of numbers ( formally, a type of subset of an ordered set )

In mathematics, a filter is a special subset of a partially ordered set.

In algebra ( which is a branch of mathematics ), a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers.

In mathematics, especially in set theory, 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.

In mathematics a topological space is called separable if it contains a countable dense subset ; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

In mathematics, a well-order relation ( or well-ordering ) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.

In mathematics, the infimum ( plural infima ) of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound ( also abbreviated as glb or GLB ) is also commonly used.

* Core ( functional analysis ), in mathematics, a subset of the domain of a closable operator

In mathematics, given a subset S of a totally or partially ordered set T, the supremum ( sup ) of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound ( lub or LUB ).

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set ( P, ≤) is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound.

In mathematics, a complete measure ( or, more precisely, a complete measure space ) is a measure space in which every subset of every null set is measurable ( having measure zero ).

In mathematics, subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.

* Closure ( mathematics ), the smallest object that both includes the object as a subset and possesses some given property

The notions of a " decidable subset " and " recursively enumerable subset " are basic ones for classical mathematics and classical logic.

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension, is a schema of axioms in Zermelo – Fraenkel set theory.

In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods.

In mathematics, logic and computer science, a formal language is called recursively enumerable ( also recognizable, partially decidable or Turing-acceptable ) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i. e., if there exists a Turing machine which will enumerate all valid strings of the language.

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U → R ( where U is an open subset of R < sup > n </ sup >) which satisfies Laplace's equation, i. e.

Reacting against authors such as J. S. Mill, Sigwart and his own former teacher Brentano, Husserl criticised their psychologism in mathematics and logic, i. e. their conception of these abstract and a-priori sciences as having an essentially empirical foundation and a prescriptive or descriptive nature.

In mathematics, Horner's method ( also known as Horner scheme in the UK or Horner's rule in the U. S .) is either of two things: ( i ) an algorithm for calculating polynomials, which consists in transforming the monomial form into a computationally efficient form ; or ( ii ) a method for approximating the roots of a polynomial.

He started studying mathematics in 1941 in the U. S., but his studies were interrupted by the war, during which he served in the military.

Quite unhappy with the lack of formal science education at Eton College, Maynard Smith took it upon himself to develop an interest in Darwinian evolutionary theory and mathematics, after having read the work of old Etonian J. B. S.

* A. S. Troelstra ( 1977a ), " Aspects of constructive mathematics ", Handbook of Mathematical Logic, pp. 973 – 1052.

Enrollment in computer-related degrees in U. S. has dropped recently due to lack of general interests in science and mathematics and also out of an apparent fear that programming will be subject to the same pressures as manufacturing and agriculture careers.

* Phillip S. Jones, Jack D. Bedient: The historical roots of elementary mathematics.

In mathematics, the symmetric group S < sub > n </ sub > on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself.

Such exploits made Knuth a topic of discussion among the mathematics department, which included Richard S. Varga.