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 interior – limit point interior   The interior of a subset E of a topological space is the largest open set contained in E. It may also be expressed as the intersection of E with the closure of its complement. If E is open, then it is equal to its interior. If E contains no open sets we say that it has empty interior, and this equivalent to saying that E is nowhere dense. intersection   The intersection of two sets A and B is the set of those elements common to both A and B, and is denoted byThus, an element x belongs to the intersection only if it is an element of A and also an element of B. If the intersection is taken over a family of sets {Ai}i = 1, 2, ... n, then it is the set of those elements that are in every set in the family, denotedSometimes one speaks of the interesection over a (single) set A, and this indicates the set of elements that are in every element of A:When considering an algebra of sets, the intersection of two or more sets is sometimes called the meet of the sets.Cf. union. interval   The set of all real numbers lying between two given real numbers a and b. If both a and b are included in the set it is called a closed interval. If neither a nor b is included it is called an open interval. If only one of a or b is included the interval is called half-open (equivalently, half-closed). Intervals may be denoted using either interval notation or set notation, as follows: Although intervals are most commonly intervals on the real line, the definition carries over without modification to any totally ordered set. interval graph   An intersection graph of a finite family of intervals on the real line. inverse image   Given a function f and a subset Y of the range of f, the inverse image (under f) of Y is the set of all x such that f(x) is in Y. inverse statement   Given a conditional statement, i.e., a statement of the form “if A then B,” or “A implies B,” its inverse is “not A implies not B.” A conditional neither implies nor is implied by its inverse. However, the converse of a conditional and its inverse are logically equivalent, since they are contrapositives of each other. irreflexive   See reflexive relation. isomorphism   A morphism that is both injective and surjective, that is, both “one-to-one” and “onto.” An isomorphism from a structure to itself is called an automorphism. isotone function   A function that is order preserving and increasing. join   A binary operation whose value on two elements a and b of a lattice is the least upper bound of a and b, denoted a b.Cf. meet. join irreducible   An element a of a lattice is called join irreducible if whenever a = b c then a = b or a = c. More generally, a is called strictly join irreducible if whenever a is the join of a subset X of the lattice, then a is an element of X.Cf. join prime, meet irreducible, meet prime. join-morphism   A function f on a lattice L is called a join-morphism if for every a and b in L we have f(a b) = f(a) f(b). If f has an inverse that is also a join-morphism, then f is called a join-isomorphism. A join-isomorphism from a lattice to itself is called a join-automorphism.Cf. join, meet-morphism. join prime   An element a of a lattice is called join prime if whenever a b c then a b or a c. Also, a is called strictly join prime ifCf. join irreducible, meet prime, meet irreducible. Jordan Decomposition Theorem   If m is a signed measure, there exist unique positive measures m+ and m– such that m = m+ – m–, with m+ and m– mutually singular. This is called the Jordan decomposition of m, and the measures m+ and m– are called the positive and negative variations of m. The total variation of m is defined to be the sum of the positive and negative variations of m. König’s Lemma   If a is an infinite cardinal and the cofinality of a is not greater than b, then ab is strictly greater than a. lattice   A partially ordered set in which each pair of elements has a least upper bound, called the join, and a greatest lower bound, called the meet. The join of two elements x and y is denoted by x y, and the meet by x y.A lattice may be equivalently defined as a set with an algebra having two binary operations (meet) and (join) defined on it, satisfyingx x = x = x x, andx (x y) = x = x (x y)A lattice is called complete if every subset of the lattice has a least upper bound and a greatest lower bound. least upper bound   An upper bound which is less than or equal to every other upper bound.Cf. greatest lower bound. least upper bound axiom   “Any subset of the real numbers which has an upper bound has a least upper bound.” This axiom, together with the field axioms, completely characterizes the set of real numbers.Cf. supremum. Lebesgue measure   The unique measure m on the real line, generated by the outer measure whose value on intervals is the length of the intervals, is called Lebesgue measure. liar paradox   According to a legend found in many ancient sources (including a letter of St. Paul), there was a certain Cretan named Epimenides who proclaimed, “All Cretans are liars.” This statement involves a contradiction on its face, since if all Cretans are liars and Epimenides is a liar, then the statement must be false, that is, not all Cretans are liars.There are many versions of the liar paradox, and many puzzles based upon it. The simplest form is, “This statement is false,” a statement that’s true if it’s false and false if it’s true. limit point   Given a sequence of points ai , i = 1, 2, 3, ... , a point L is called a limit point of the sequence if every neighborhood of L contains all but finitely many of the ai .If X is a metric space, then L is a limit point of X if a sequence from X may be chosen so that L is the limit point of that sequence. Limit points are not in general unique.See also: accumulation point, perfect set. interior – limit point
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