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If x is a member of a set A, then x is said to be an element of A, written x in A. If x is not an element of A, this is written x not in A. The term element also refers to a ...

A set A in a first-countable space is dense in B if B=A union L, where L is the set of limit points of A. For example, the rational numbers are dense in the reals. In ...

The ascending chain condition, commonly abbreviated "A.C.C.," for a partially ordered set X requires that all increasing sequences in X become eventually constant. A module M ...

A topological basis is a subset B of a set T in which all other open sets can be written as unions or finite intersections of B. For the real numbers, the set of all open ...

Given a set P of primes, a field K is called a class field if it is a maximal normal extension of the rationals which splits all of the primes in P, and if P is the maximal ...

Let a graph G=(V,E) be defined on vertex set V and edge set E. Then a construction sequence (or c-sequence) for G is a linear order on V union E in which each edge appears ...

The mode of a set of observations is the most commonly occurring value. For example, for a data set (3, 7, 3, 9, 9, 3, 5, 1, 8, 5) (left histogram), the unique mode is 3. ...

The set R union {infty}, obtained by adjoining one improper element to the set R of real numbers, is the set of projectively extended real numbers. Although notation is not ...

The supremum is the least upper bound of a set S, defined as a quantity M such that no member of the set exceeds M, but if epsilon is any positive quantity, however small, ...

The Cartesian product of two sets A and B (also called the product set, set direct product, or cross product) is defined to be the set of all points (a,b) where a in A and b ...