circumference complex number
Geometry: The distance around a circle in the plane, or around a great circle of a sphere.
Graph Theory: The circumference of a graph G is defined as the length of the longest cycle of G. The circumference is ususally denoted by c(G), and is undefined if G has no cycles.
See proper class.
General: A set is closed under an operation if applying the operation to its elements returns only elements in the set. For example, the set of integers is closed under addition, since adding two integers always gives another integer, but it is not closed under division, since dividing two integers may result in a non-integer.
Geometry: A plane figure is closed if it consists of lines and/or curves that entirely enclose an area. Similarly, a figure in 3-dimensional space is called closed if it entirely encloses a volume.
See the following listings for other uses of the word “closed” in mathematics.
Topology: A set is topologically closed if it is not open.
An interval of the real number line (or any other totally ordered set) which includes its endpoints. An interval containing only one of its endpoints is called half-open.
Cf. open interval.
Topology: A subset E of a topological space X is closed if X - E (set difference) is open. In a metric space, E is closed if every convergent sequence in E converges in E; equivalently, if every accumulation point of E is in E.
Set Theory: If a is a limit ordinal, then a set C contained in a is called closed if and only if for every limit ordinal b less than a, if C b is unbounded in b, then b C. C is called c.u.b. (“cub set” or “club set”) if and only if C is closed and unbounded in a.
Cf. stationary set.
closed set system
If X is a set (or proper class) and F is a family of subsets of X, then F is called a closed set system provided
- X is a member of F, and
- F is closed under arbitrary intersections.
If X is a set, then a function C from P(X) into P(X) (i.e., a function on the power set of X) is called a closure operator provided
Closure operators induce closed set systems.
- Y is contained in C(Y) for every subset Y of X,
- C(C(Y)) = C(Y) for every subset Y of X, and
- If Y and Z are both subsets of X, with Y a subset of Z, then C(Y) is a subset of C(Z).
A property of subsets of a set X is a closure property if X has the property and the intersection of any subsets of X having the property also has the property.
Closed, unbounded set. See closed set.
A function f which maps an ordinal a into an ordinal b is said to map a cofinally if the range of f is not bounded in b. (I.e., for every b in b there is an a in a such that f(a) b.) The cofinality of an ordinal b is the least ordinal a such that there is a cofinal map of a into b.
A subselection of a set of r elements from a set of n elements. The number of such combinations, i.e., the number of ways in which r elements may be chosen from a set of n elements, is given by the formula
This operation is also sometimes denoted by nC r, and is read “n choose r.”
Cf. permutation, factorial.
A logarithm with base 10.
An operation “ · ” on elements of a set A is commutative if for all elements a, b in A, a · b = b · a.
Cf. associative, distributive property.
A property of numbers which states that the operations of addition and multiplication are commutative. Cf. distributive property, associative.
The complement of a set A is the set of all elements that are not elements of A.
Graph Theory: The complement of a simple graph G with vertex set V is the simple graph Gc, which also has vertex set V, and in which two vertices are adjacent if and only if they are not adjacent in G.
Two angles are complementary if they add up to a right angle.
Analysis: A metric space X is complete if every Cauchy sequence in X converges in X.
Logic: a system of axioms for a mathematical theory is complete if every theorem in the theory is deducible from the axioms. Gödel's incompleteness theorem states that any axiom system which includes or allows the operations of arithmetic is necessarily incomplete.
Set Theory: If F is a filter on a set X and k is a regular, uncountable cardinal, then we say that F is k-complete (k-closed) if AF for every AF with |A| < k. Every filter is w-complete. If k is the first uncountable cardinal (1), then F is called countably complete.
A lattice X is complete if every subset of X has a least upper bound and a greatest lower bound in X.
complete ordered field
A linearly ordered field in which every subset with an upper bound has a least upper bound. Every complete ordered field is isomorphic to the set of real numbers.
An element of the set C of numbers of the form a + bi, where a and b are real numbers and i 2 = -1. The number i is called the imaginary number.