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bijection – Cauchy sequence
bijection
A bijective function, i.e., a function that is both an injection and a surjection.
bijective
A function is bijective if it is both injective and surjective, i.e., both “one-to-one” and “onto.”
binary operation
A binary operation on a set X is a function whose domain is the set of ordered pairs of elements from X and whose range is X.
Examples: addition on the set of integers, function composition.
Bolzano-Weierstrass property
A topological space X is said to have the Bolzano-Weierstrass property if every infinite sequence in X has a convergent subsequence. It is a theorem that every compact space has the Bolzano-Weierstrass property.
Borel measure
A measure m on a metric space X defined on a s-algebra of sets which contains the Borel sets is called a Borel measure. See also: Lebesgue measure.
Borel set
If X is a topological space, then the s-algebra of sets generated by the open sets of X is called the Borel s-algebra, and its members are called Borel sets. The Borel subsets of the real line are generated by the open intervals (equivalently, by the closed intervals, half-open intervals, and rays).
bound
A lower bound of any subset of a linear order (linearly ordered set) is an element which is less than or equal to every element of the subset. The greatest lower bound is the largest of its lower bounds. An upper bound of any subset of a linear order is an element which is greater than or equal to every element of the subset. The least upper bound is the smallest of the upper bounds.
Cf. infimum, supremum.
bounded
A set or sequence of values is called bounded if there is a value M such that the values are never bigger than M and never smaller than –M. A function is bounded if its values are bounded. A sequence of functions is pointwise bounded if at every domain element x the sequence of values of the functions at x is bounded. A subset E of a locally compact topological space is bounded if there exists a compact set C such that E is contained in C. Such a subset is called s-bounded if there exists a sequence of compact sets Ci such that E is contained in their union. See also: totally bounded.
bounded variation
A function on the real numbers with range in the complex numbers is said to be of bounded variation if its total variation is finite. The space of all such functions is denoted by BV. If in addition f is right continuous and its limit at negative infinity is zero, then f is said to be of normalized bounded variation. The space of such functions is denoted by NBV.
Cantor-Bendixson Theorem
Every closed subset of the real numbers is the disjoint union of a perfect set and a countable set.
Cf. derived set.
Cantor-Schröder-Bernstein Theorem
For any sets A and B, if there exists an injective (one-to-one) function from A into B and also an injective function from B into A, then there exists a surjective (onto) function from A onto B.
Cantor set
ARTICLE
A subset of the unit interval (on the real number line) which is a perfect set, nowhere dense, uncountable, and homeomorphic to the entire unit interval. See the article for a complete exposition.
Cantor’s Theorem
ARTICLE
Given any set, a new set may be constructed which is cardinally larger, i.e., whose size is represented by a larger cardinal than the size of the initial set. More specifically, for no set X is there a bijection of X onto its power set. This theorem establishes both that there are different sizes of infinity, and that there is no greatest such “size.” Cantor’s proof was the first use of a diagonalization argument to derive a contradiction.
Related MiniText: Infinity -- You Can't Get There From Here...
Carathéodory’s Theorem
See: outer measure.
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
Carroll’s Paradox
ARTICLE
Paradox published by Lewis Carroll as “What the Tortoise Said to Achilles.” See the article for a full exposition.
Cf. Charles Lutwidge Dodgson.
Cartesian plane
The Cartesian product R2, represented graphically by two real number lines at right angles to one another, with the point (0,0) at the intersection.
Used for graphing functions from the set of real numbers to itself. The quadrants, numbered I - IV as shown, indicate the regions of the plane where the x and y axes are positive and negative. Compare: Argand plane.
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.
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