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cardinal – convergent sequence
cardinal
A cardinal number represents the size of a set irrespective of the order or structure of its elements. If X is a set, its cardinality is generally indicated by |X|. Two sets A and B are said to have the same cardinality if there is a function from A into B that is bijective (i.e., one-to-one and onto).
 A cardinal is an initial ordinal, i.e., an ordinal for which there does not exist a bijection onto any lesser ordinal. Thus, all finite ordinals (i.e., the natural numbers) are cardinals, but most transfinite ordinals are not cardinals.
If a is a cardinal, then we denote by a+ the least cardinal greater than a. A cardinal k is called a successor cardinal if k = a+ for some cardinal a. Otherwise k is called a limit cardinal. The heirarchy of transfinite cardinals is defined by recursion, and denoted using the Hebrew letter aleph:
where w is the first infinite ordinal.
Cf. cardinal arithmetic, inaccessible cardinal, regular.
Related MiniText: Infinity -- You Can't Get There From Here...
cardinal arithmetic
Let k and l denote cardinals and A and B denote sets such that k = |A| and l = |B|. Denote by BA the set of all functions from B to A, and by A × B the Cartesian product of A and B. The operations of cardinal arithmetic are then defined by- k + l = |A union B|;
- k • l = |A × B|;
- kl = |BA|
For finite cardinals, this corresponds to the operations of ordinary arithmetic, but is quite different in the case of transfinite cardinals. For instance, if k is not finite, then k + k = k.
Cf. ordinal arithmetic.
cardinality
Two sets X and Y have the same cardinality if there exists a bijection between them. Specifically, the cardinality of X may be understood as a canonical object meant to represent the proper class of sets to which X is bijective. If the axiom of choice is assumed, then the cardinality of X is the unique cardinal number having the same cardinality as X
Cartesian product
For any collection {Ai}, i = 1, 2, 3, ..., n, of sets, the Cartesian product
is 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 that
In 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.
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.
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 provided- X is a member of F, and
- F 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 provided- 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).
Closure operators induce closed set systems.
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:
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 A F 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
complete measure
See measure.
concave
A region of space is concave if there are two points of the region such that a line joining the two points is not entirely contained within the region. In particular, a polygon is concave if any of its interior angles is greater than 180°.
Cf. convex.
concave function
A function is concave if the chord connecting any two points of its graph lies entirely below the graph.
Cf. convex function.
connected
A topological space X is connected if there are no two open sets of X whose union is X and whose intersection is empty.
constant function
A constant function f is one whose value is the same at all points of its domain.
continuous
Analysis: A function f is continuous at a point x of its domain if, whenever we are given a number e greater than 0, we may find a d greater than 0 so that whenever y is within a d-neighborhood of x, then f(y) is within an e-neighborhood of f(x).
Topology: A transformation of one topological space into another is continuous if the inverse image of every open set is open, or equivalently if the inverse image of every closed set is closed.
Cf. uniformly continuous, equicontinuous, absolutely continuous.
convergent sequence
See sequence.
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