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neighborhood – order type
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.
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.
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.
open covering
In a topological space, an open covering of a set E is a collection {Ui} of open sets such that E is contained in the union of the Ui.
open disk
The interior of a circle.
Cf. neighborhood, disk.
open function
A function from one topological space into another is called open if the image of every open set of the domain is an open set in the range.
open interval
An interval of the real number line (or any other totally ordered set) which does not include its endpoints. An interval containing only one of its endpoints is called half-open.
Cf. closed interval.
open set
A subset U of a topological space X is open if every element x of U is contained in an open set of X that is also contained in U. In a metric space, U is open if for every x in U we may find a d greater than zero such that the d neighborhood of x is also contained in U.
ordered field
See field.
ordered pair
An ordered tuple (a,b), the first element of which is called the abscissa, and the second element the ordinate, and for which (a,b) = (b,a) if and only if a = b. Functions, graphs of functions, and binary relations are represented as sets of ordered pairs. In standard set theory, the ordered pair (a,b) is defined to be the set { {a}, {a,b} }.
Cf. flat pair.
ordered set
A set with an order relation defined on it.
Cf. partial order, total order.
order-preserving function
A function f is called order-preserving if it preserves the order of its domain elements, that is, if whenever x and y are elements of its domain such that x  y then f(x) f(y). Also called isotone or inctreasing. If f reverses the order of its domain elements, then it is called antitone or decreasing. In either case f is called monotone or monotonic. If whenever x < y we have f(x) < f(y), then f is called strictly increasing (resp. decreasing).
order relation
A relation R on a set S is an order relation exactly if it is reflexive, transitive and antisymmetric. Order relations are usually denoted by “ < ” or “ ”.
Cf. partial order, total order
order topology
A topology on a totally ordered set that agrees with the order. Specifically, given a totally ordered set X with total order relation <, we define the order topology T on X to be the collection of all arbitrary unions of open intervals of X under <.
order type
See total order.
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