BROWSE ALPHABETICALLY LEVEL:    Elementary    Advanced    Both INCLUDE TOPICS:    Basic Math    Algebra    Analysis    Biography    Calculus    Comp Sci    Discrete    Economics    Foundations    Geometry    Graph Thry    History    Number Thry    Phys Sci    Statistics    Topology    Trigonometry Cartesian product – complete Cartesian product   For any collection {Ai}, i = 1, 2, 3, ..., n, of sets, the Cartesian productis the set of ordered n-tuples (a1,a2, ... ,an) with a1 an element of A1, a2 an element of A2, etc. The Cartesian product R2 of the set of real numbers is called the Cartesian plane, and in general n-dimensional real space is the Cartesian product Rn. The assertion that the Cartesian product of an infinite collection of non-empty sets is non-empty is equivalent to the axiom of choice. Cauchy sequence   A sequence (x1, x2, x3, ... ) of elements of a metric space X with metric d(x, y) is Cauchy if for any e greater than zero there is some natural number N such thatIn other words, in a Cauchy sequence, the elements eventually become “arbitrarily close together.” If the metric space X is closed, this condition is equivalent to the sequence being convergent. CH   See: continuum hypothesis. chain   If X is a partially ordered set, then a subset Y of X is called a chain if it is totally ordered, that is, if for any two elements a, b of Y, either a b or b a.Cf. antichain. chain condition   For a an infinite cardinal, a partial order P is said to have the a-chain condition if every antichain in P has cardinality not greater than a. If a = w, this is called the countable chain condition, or “c.c.c.” characteristic function   Given a subset E of a space X, the characteristic function cE is defined by cE(x) = 1 if x is in E, and cE(x) = 0 otherwise. All properties of sets and set operations may be expressed by means of characteristic functions. choice, axiom of   See: axiom of choice. circumference   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. class   See proper class. closed interval   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. closed set   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 providedX is a member of F, andF is closed under arbitrary intersections.Cf. filter. closure   Topology: The closure of a subset E of a topological space is the smallest closed set containing E. It may also be expressed as the union of E with its accumulation points. If E is closed, then it is equal to its closure.Algebra: An algebraic closure of a field F is a field G containing F such that every polynomial with coefficients from F has a root in G. closure operator   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 providedY is contained in C(Y) for every subset Y of X, C(C(Y)) = C(Y) for every subset Y of X, andIf Y and Z are both subsets of X, with Y a subset of Z, then C(Y) is a subset of C(Z). Closure operators induce closed set systems. closure property   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. club set   Closed, unbounded set. See closed set. cofinality   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. compact   Topology: In a topological space, a set E is compact if every open covering of E has a finite subcover, i.e., a finite subcollection which also covers E. A space X is compact if and only if every collection of closed sets with the finite intersection property has a non-empty intersection. E is called s-compact if there exists a sequence of compact sets {Ci} such that E is contained in their union.Cf. locally compact, Bolzano-Weierstrass property, Heine-Borel property.Set Theory: A cardinal k is called weakly compact if it is uncountable and Equivalently, k is weakly compact if it is strongly inaccessible and there are no k-Aronszajn trees.Lattices: an element a of a lattice L is called compact if whenever a is dominated by the join of a subset X of L then a is dominated by the join of a finite subset of X. Symbolically: comparison test    ARTICLE   A test for the convergence of a series. See the article for a complete description. complement   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. complete   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. Related article: Gödel's Theorems Cartesian product – complete
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