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metric space – null set
metric space
A set with a metric defined on its elements.
monotone function
Also called monotonic function. See order-preserving function.
morphism
A function from one set to another is called a morphism if it preserves some designated structural properties or operations on the domain set. Typically, the word morphism is not used by itself, but in combination with a prefix that indicates whether it is injective, surjective, etc.
Cf. automorphism, epimorphism, homeomorphsim, homomorphism, isomorphism, monomorphsim.
multiplication
A binary operation on numbers or quantities resulting in a product, usually but not always amounting to repeated addition. On the natural numbers multiplication is defined recursively by the Peano axioms, such that the product of two numbers n and m, denoted by n × m, is found by adding up m copies of n (or n copies of m).
Multiplication of most kinds of numbers is associative and commutative, but these properties sometimes fail, for example in the case of matrix multiplication.
When a product of more than two numbers or quantities is taken, the general product may be denoted by the capital Greek letter Pi, i.e., Pai denotes the product a1 × a2 × . . . × an.
multiply
To find the product of two numbers or quantities by multiplication.
mutually prime
Two integers are mutually prime if they have no common factors larger than 1 or -1.
mutually singular
Given a measurable space (X, M), two signed measures m and n on X are called mutually singular, denoted by
if there exist sets E and F in M such that their union is X, their intersection is empty, and m(E) = 0 and n(F) = 0.
natural base
See Euler number.
natural logarithm
A logarithm with base e, the Euler number. Often written “ln” rather than “log” to distinguish it from logarithms using other bases.
natural number
An element of the set N = {1, 2, 3, ...} consisting of all the “counting numbers.” When the number 0 is included, this set is sometimes called the whole numbers. In set theory, the natural numbers (incuding 0) are identified with the set w of finite ordinals. The natural numbers are a well-founded linear order with no largest member, and are countably infinite.
Cf. Peano axioms, rational number, real number.
Related MiniText: Number -- What Is How Many?
negation
If j is a statement, sentence, or formula of logic, then the negation of j, denoted by  j, is that formula which is true whenever j is false, and false whenever j is true.
negative
The negative of a number or quantity x is the number, denoted -x, which when added to x yields 0. That is, the negative of a number is its additive inverse.
negative set
Given a signed measure m on a measure space X, a measurable set A in X is called a negative set if the measure of all measurable subsets of A is less than or equal to zero.
Cf. positive set, null set.
neighborhood
A neighborhood of a point x of a topological space is an open set of the space containing x. In a metric space, a d-neighborhood of x is the collection of all points of the space whose distance from x is less than d.
non-denumerable
Uncountable.
norm
Analysis: A non-negative real-valued function “|| x ||” defined on a vector space, satisfying- || –x || = || x ||,
- || cx || = || c || × || x || for all scalars c, and
- || x + y || <= || x || + || y || (triangle inequality)
Statistics: Another term for the mode of a frequency distribution.
normal
A line intersecting a curve (or surface) perpendicular to the tangent line (or tangent plane) at the point of intersection. The normal to a surface expressed as a function of several variables xi is given by the gradient.
normalized bounded variation
See: bounded variation.
normed space
A vector space with a norm defined on it.
nowhere dense
Given a space X and a subset A of X, we say that A is nowhere dense if every open set of X contains an open subset that is disjoint from A. This is equivalent to saying that the complement of A is dense, or that A has empty interior.
null set
A set of measure zero. That is, given a measure m on a measure space X, a measurable set A in X is called a null set if its measure is zero.
Cf. positive set, negative set, almost everywhere.
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