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neighborhood – pointwise bounded
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.
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 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.
outer measure
A non-negative extended real-valued set function defined on all subsets of a space X that is zero on the empty set, monotonic, and countably subadditive (see below) is called an outer measure. An outer measure is often used together with Caratheodory's Theorem (see below) to obtain a measure. Given a set X and a collection A of subsets of X which includes the empty set and X itself, and a positive real-valued function r whose domain is A and whose value on the empty set is zero, then for any F in X define m* by
Then m* is an outer measure. A set B in X is then called m*-measurable if
for all subsets C of X. (Cc denotes the complement of C in X.) Carathéodory’s Theorem states that if m* is an outer measure on X, then the collection M of m*-measurable sets is a s-algebra of sets, and the restriction of m* to M is a complete measure.
partition
General: A partition of a set X is a collection of subsets of X such that every element of X is in exactly one of the subsets. Such a partition is given by (and gives rise to) an equivalence relation on X. For example, division modulo 3 partitions the set of natural numbers into three subsets, each containing all those numbers leaving remainders of 0, 1, or 2 respectively when divided by 3.
Algebra: A partition of a matrix is a division of the matrix into conformable submatrices.
Analysis: A partition of a space is a collection of pairwise disjoint regions of the space whose union is the entire space. For example, a partition of an interval [a, b] of the real line is given by a finite set of points {xi} such that a = x1 < x2 < . . . < xn = b which divide the interval into disjoint subintervals.
Number Theory: Given a positive integer n, a partition of n is a set of positive integers whose sum is n. For example, 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1 are the four possible partitions of the number 4.
perfect set
A closed set X is called perfect if every point of X is an accumulation point of X. The following are equivalent characterizations:- X is closed and containes no isolated points.
- X is closed and dense in itself.
- X is equal to its derived set.
pointwise bounded
See bounded.
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