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 morphism – ordered field morphism   A function from one set to another is called a morphism if it preserves some designated structural properties or operations on the domain set. Typically, the word morphism is not used by itself, but in combination with a prefix that indicates whether it is injective, surjective, etc.Cf. automorphism, epimorphism, homeomorphsim, homomorphism, isomorphism, monomorphsim. 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. 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 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. 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. 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. morphism – ordered field
 HOME | ABOUT | CONTACT | AD INFO | PRIVACYCopyright © 1997-2013, Math Academy Online™ / Platonic Realms™. Except where otherwise prohibited, material on this site may be printed for personal classroom use without permission by students and instructors for non-profit, educational purposes only. All other reproduction in whole or in part, including electronic reproduction or redistribution, for any purpose, except by express written agreement is strictly prohibited. Please send comments, corrections, and enquiries using our contact page.