The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory.

It is also commonly used in mathematics in algebraic solutions representing quantities such as angles.

Associative operations are abundant in mathematics ; in fact, many algebraic structures ( such as semigroups and categories ) explicitly require their binary operations to be associative.

In his youth he went to the continent and taught mathematics in Paris, where he published or edited, between the years 1612 and 1619, various geometric and algebraic tracts.

Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution ; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.

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

Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact.

Even though equivalence relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders.

In mathematics and abstract algebra, a group is the algebraic structure, where is a non-empty set and denotes a binary operation called the group operation.

In modern mathematics, the theory of fields ( or field theory ) plays an essential role in number theory and algebraic geometry.

In mathematics, more specifically algebraic topology, the fundamental group ( defined by Henri Poincaré in his article Analysis Situs, published in 1895 ) is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

( See areas of mathematics and algebraic geometry.

" Kronecker even objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum.

Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing.

In mathematics, a Lie algebra (, not ) is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group ; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi ; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

The development of algebraic geometry from its classical to modern forms is a particularly striking example of the way an area of mathematics can change radically in its viewpoint, without making what was correctly proved before in any way incorrect ; of course mathematical progress clarifies gaps in previous proofs, often by exposing hidden assumptions, which progress has revealed worth conceptualizing.

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element.

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.

For example, rather than just getting a passing grade for mathematics, a student might be assessed as level 4 for number sense, level 5 for algebraic concepts, level 3 for measurement skills, etc.

In advanced mathematics, polynomials are used to construct polynomial rings, a central concept in abstract algebra and algebraic geometry.

In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that " division " is always possible.

* Ring ( mathematics ), an algebraic structure

Connes has applied his work in areas of mathematics and theoretical physics, including number theory, differential geometry and particle physics.

One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up.

In mathematics, the absolute value ( or modulus ) of a real number is the numerical value of without regard to its sign.

In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

In mathematics, the phrase " almost all " has a number of specialised uses.

It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

Realism in the philosophy of mathematics is the claim that mathematical entities such as number have a mind-independent existence.

Arithmetic or arithmetics ( from the Greek word ἀριθμός, arithmos " number ") is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations.

* Bell number, in mathematics

In mathematics, the Bernoulli numbers B < sub > n </ sub > are a sequence of rational numbers with deep connections to number theory.

With large sets, it becomes necessary to use more sophisticated mathematics to find the number of combinations.

* Catalan number, a concept in 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.

Although a " bijection " seems a more advanced concept than a number, the usual development of mathematics in terms of set theory defines functions before numbers, as they are based on much simpler sets.

In 1879, Peirce was appointed Lecturer in logic at the new Johns Hopkins University, which was strong in a number of areas that interested him, such as philosophy ( Royce and Dewey did their PhDs at Hopkins ), psychology ( taught by G. Stanley Hall and studied by Joseph Jastrow, who coauthored a landmark empirical study with Peirce ), and mathematics ( taught by J. J. Sylvester, who came to admire Peirce's work on mathematics and logic ).

The Langlands program is a far-reaching web of these ideas of ' unifying conjectures ' that link different subfields of mathematics, e. g. number theory and representation theory of Lie groups ; some of these conjectures have since been proved.

In mathematics, any number of cases supporting a conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample would immediately bring down the conjecture.

* Cardinal number, a concept in mathematics

In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties.

In mathematics, the cardinality of a set is a measure of the " number of elements of the set ".

In mathematics, a contraction mapping, or contraction, on a metric space ( M, d ) is a function f from M to itself, with the property that there is some nonnegative real number < math > k < 1 </ math > such that for all x and y in M,