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nowhere dense – order topology
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.
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 of operations
As a matter of convention, in any given expression involving arithmetic and/or algebraic operations, operations within parentheses (or other grouping symbols) are evaluated first. Within this constraint, exponentiation precedes multiplication and division, and the latter precede addition and subtraction. Within these constraints, operations are evaluated from left to right.
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 <.
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