BROWSE ALPHABETICALLY LEVEL:    Elementary    Advanced    Both INCLUDE TOPICS:    Basic Math    Algebra    Analysis    Biography    Calculus    Comp Sci    Discrete    Economics    Foundations    Geometry    Graph Thry    History    Number Thry    Phys Sci    Statistics    Topology    Trigonometry metric space – open set metric space   A set with a metric defined on its elements. monotone function   Also called monotonic function. See order-preserving function. mutually singular   Given a measurable space (X, M), two signed measures m and n on X are called mutually singular, denoted byif 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. nth-term test   A test for the divergence of a series. See the related article for a complete description. Related article: Series natural logarithm   A logarithm with base e, the Euler number. Often written “ln” rather than “log” to distinguish it from logarithms using other bases. 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   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. 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 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. metric space – open set
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