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, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are " 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.

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.

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

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.

In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X

In mathematics, the axiom of regularity ( also known as the axiom of foundation ) is one of the axioms of Zermelo – Fraenkel set theory and was introduced by.

The axiom of regularity is arguably the least useful ingredient of Zermelo – Fraenkel set theory, since virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity ( see chapter 3 of ).

This is philosophically unsatisfying to some and has motivated additional work in set theory and other methods of formalizing the foundations of mathematics such as New Foundations by Willard Van Orman Quine.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A.

In mathematics, a binary operation on a set is a calculation involving two elements of the set ( called operands ) and producing another element of the set ( more formally, an operation whose arity is two ).

Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics.

The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics ( for example Venn diagrams and symbolic reasoning about their Boolean algebra ), and the everyday usage of set theory concepts in most contemporary mathematics.

The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo – Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent.

Certain categories called topoi ( singular topos ) can even serve as an alternative to axiomatic set theory as a foundation of mathematics.

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 mathematics, specifically general topology and metric topology, a compact space is a mathematical space in which any infinite collection of points sampled from the space must — as a set — be arbitrarily close to some point of the space.

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, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of X ; this is known as the σ-algebra of Borel sets.

A surjective function from domain ( mathematics ) | domain X to codomain Y.

In mathematics, a function f from a set X to a set Y is surjective ( or onto ), or a surjection, if every element y in Y has a corresponding element x in X so that f ( x ) = y.

A non-surjective function from domain ( mathematics ) | domain X to codomain Y.

* José Ferreirós, José Ferreirós Domínguez, Labyrinth of thought: a history of set theory and its role in modern mathematics, Edition 2, Springer, 2007, ISBN 3-7643-8349-6, chapter X " Logic and Type Theory in the Interwar Period "

In mathematics, the convex hull or convex envelope of a set X of points in the Euclidean plane or Euclidean space is the smallest convex set that contains X.

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T < sub > 4 </ sub >: every two disjoint closed sets of X have disjoint open neighborhoods.

In mathematics, a partition of unity of a topological space X is a set of continuous functions,, from X to the unit interval such that for every point,,

In mathematics, a base ( or basis ) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B.

In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X.

In topology and related branches of mathematics, a topological space X is a T < sub > 0 </ sub > space or Kolmogorov space if for every pair of distinct points of X, at least one of them has an open neighborhood not containing the other.

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.

* Lift ( mathematics ), a morphism h from X to Z such that gh = f

She is best known for her role as Winnie Cooper in the television show The Wonder Years, and later as the New York Times bestselling author of four popular non-fiction books: Math Doesn't Suck, Kiss My Math, Hot X: Algebra Exposed and Girls Get Curves: Geometry Takes Shape, which encourage middle-school and high-school girls to have confidence and succeed in mathematics.