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natural number – open cover
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.
nonagon
A regular polygon having nine equal sides and nine equal angles.
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.
normal subgroup
See subgroup.
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.
number
There is no precise mathematical definition of the word “number.” There are however precise definitions of the terms “natural number,” “rational number,” “real number,” “complex number,” and other less commonly used kinds of number. When a mathematician speaks about numbers she usually has one of these cases in mind and she should, at the outset, make it clear to which type of number she is referring. The naive, inborn concept of number that is shared to some degree by all humans is a matter for philosophical rather than strictly mathematical inquiry, and it may be noted that there has historically been strong opposition to the introduction of new generalizations of established concepts of number.
numeral
Graphical symbol representing a number.
obtuse
An angle is called obtuse if it is greater than a right angle, that is, if its measure is greater than 90° (p/2 radians). A triangle is called obtuse if one of its angles is obtuse.
Cf. acute.
octahedron
A polyhedron having eight faces.
The faces of a regular octahedron are congruent, equilateral triangles.
Cf. Platonic solid, polyhedron.
odd function
A real-valued function y = f(x) is odd if f(–x) = –f(x) for all x in the domain of f. The graphs of odd functions in the Cartesian plane are symmetric with respect to the origin.
Cf. even function.
open
See: open function, open interval, open set.
open cover
A collection of open sets which contains a given set X is called an open cover of X.
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