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surjective – well-ordering principle
surjective
A function f from a set X to a set Y is surjective, also called “onto,” if to each element y of Y there is an element x of X such that f maps x to y, i.e., f (x)=y. Compare: injective, bijective.
Taylor series
Given a function having derivatives of all orders, the Taylor series of the function is given by
where f (k)(a) is the kth derivative of f at a. A function is equal to its Taylor series if and only if its error term Rn can be made arbitrarily small, where
It can be shown that
for some c between a and x.
topological space
A set with a topology defined upon it.
topology
Generally, topology is the study of those properties of a space which are invariant under continuous deformations, i.e., deformations which do not create “tears” or “holes.” More specifically, given a set X, a topology on X is a collection of subsets of X, called the open sets of X, such that the empty set and X itself are included in the collection, and such that the collection is closed under the formation of finite intersections and arbitrary (i.e., not necessarily finite or countable) unions. A set X with a topology defined upon it is called a topological space.
Cf. homeomorphism.
totally bounded
Given a metric space X, a subset E of X is called totally bounded if for every e greater than zero there is a finite covering of E by open spheres in X whose radius is less than e.
Cf. bounded.
totally ordered set
A set with a total order defined on it.
total variation
(For total variation of a signed measure, see the Jordan Decomposition Theorem.) If f is a function on the real numbers with range in the complex numbers, then the function Tf given by
is called the total variation function of f, where here the supremum is being taken over all finite partitions of the real line up to x. If the limit of this function as x goes to infinity is finite, then f is said to be of bounded variation. The space of all such functions is usually denoted by BV. Functions which are increasing and bounded are in BV, and differentiable functions whose derivative is bounded are in BV.
transcendental function
A function which is not an algebraic function, i.e., a function whose action on its argument(s) cannot be represented by the arithmetic and algebraic operations: addition and subtraction, multiplication and division, raising to a power, or extraction of roots. The exponential function, the logarithmic function, and the trigonometric functions are examples of transcendental functions.
trivial topology
For a given space X, the trivial topology is the topology whose only open sets are the empty set and X itself.
Tychonoff’s Theorem
If {Xa}, a in some index set A, is any family of compact topological spaces, then the cartesian product of the Xa, with the product topology, is compact.
uncountable
A set is uncountable (uncountably infinite) if it is infinite but not countable, i.e., no complete one-to-one match-up of the set with the set of natural numbers (finite ordinals) can be performed. Georg Cantor proved that the set of real numbers is uncountably infinite. (This is sometimes called the “non-denumerability of the continuum”).
Related MiniText: Infinity -- You Can't Get There From Here...
uncountably infinite
See uncountable.
unit circle
A circle with radius 1.
unit interval
The interval on the real number line from 0 to 1, inclusive.
unit square
The set of points of the Cartesian plane with domain and range values in the unit interval, that is the square region with vertices (0, 0), (0, 1), (1, 0), and (1, 1), including its boundary.
upper bound
An upper bound of a set with an order relation (such as “ < ”) defined on it is an element which is greater than or equal to every element in the set.
Cf. least upper bound.
vector
A quantity having two components; a magnitude component and a direction component. In n-dimensional Euclidean space, a vector is representable by an ordered tuple (a1, a2, a3, ... an) whose elements are called the components of the vector. In this case the magnitude of the vector is given by the square root of the sum of the squares of the components of the vector.
vector product
The vector product (also called cross product) of two vectors u and v, denoted u × v and called “u cross v,” is a vector w whose magnitude (length) is the product of the magnitudes of u and v and the sine of the angle between them, and which points in a direction perpendicular to the plane containing u and v so as to form a right-handed system, as in the figure.
Note that the directedness of the vector product implies that it is not commutative.
Cf. scalar product.
vector space
A structure consisting of two kinds of elements called scalars and vectors, with operations of addition of pairs of scalars or pairs of vectors, and multiplication of pairs of scalars or a scalar and a vector. The vectors form an Abelian group under addition, and the scalars form a field under their operations, and the vector space is said to be over that field.
If the scalar field is the real numbers or the complex numbers and the vectors are in n-dimensional real or complex space, then the space is called an n-dimensional real or complex vector space accordingly. Multiplication of vectors by scalars is associative with scalar multiplication, and distributive over both scalar addition and vector addition. Symbolically, for scalars a, b, and vectors u, v,
Vector spaces are usually denoted by V, and it is conventional to write the scalar on the left of a scalar multiplication. When there is any possibility of confusion, the vectors of a vector space are usually specially marked, either by drawing a (right pointing) arrow over them or by writing them in bold face.
Cf. module.
well-ordered
A set S with a linear order is called well-ordered if every non-empty subset T of S has a least element under the ordering relation.
Cf. well-ordering principle.
well-ordering principle
The assertion that every set can be well-ordered. Equivalent to the Axiom of Choice.
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