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 open set – Stone-Weierstrass Theorem 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 iffor 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. positive set   Given a signed measure m on a measure space X, a measurable set A in X is called a positive set if the measure of all measurable subsets of A is greater than or equal to zero.Cf. negative set, null set. precompact   Given a topological space X, a subset E of X is called precompact if its closure is compact. p-series   An infinite series of the formwith p a positive real number. See the related article for details. Related article: Series regular measure   A Borel measure m on a finite-dimensional real space is called regular if for all compact measurable sets K and Borel sets E we have: Riemann Hypothesis   The conjecture that the zeta function has no non-trivial zeros off of the line Re(z) = 1/2. Riemann integral   See integral. Riemann sum   Let f be a real-valued function defined on the closed interval [a, b], and let D be a partition of [a, b], i.e., a = x0 < x1 < ... < xn = b, and where Dxi is the width of the i th subinterval. If c i is any point in the i th subinterval, then the sumis called the Riemann sum of f for the partition D. semi-finite measure   See measure. separable   A topological space is separable if it has a countable base. series    ARTICLE   A series is an infinite sum, where the nth summand is the nth term of a sequence. A series is usually denoted using “sigma notation,” i.e.,The index n may begin with 0, 1, or k for any natural number k, as a matter of convenience. The nth partial sum Sn of a series is the (finite) sum of the first n terms of the series. A series is said to converge if and only if its sequence of partial sums {S 1, S 2, . . . , Sn, . . . } is a convergent sequence. There are several important types of series and several tests for the convergence of a series. Additionally, most useful functions have Taylor series representations, which makes them very important in the study of differential equations. See the article for a complete description. signed measure   Given a set X together with a s-algebra of sets M defined on it, a signed measure on (X, M) is an extended real-valued function m with domain M satisfying: The signed measure of the empty set is zero.The signed measure m assumes at most one of the values +/- infinity.(Countable additivity) Given a countable sequence of disjoint sets in M, the signed measure of the union of the sequence is equal to the sum of the signed measures of the sets in the sequence, where this sum converges absolutely if the signed measure of the union is finite. Technically speaking, every measure is a signed measure; ordinary (i.e., nowhere negative) measures are sometimes called positive measures. sphere   A closed surface, all points of which are equidistant from a given point, called the center. In 3-dimensional Euclidean space, the equation of a sphere of radius r and center (h, j, k) isThe term sphere may also refer to the solid bounded by this surface, and the interior is then called the open sphere of radius r.More generally, a sphere may be defined as the set of points in n-dimensional space (or any metric space) equidistant from a given point. The unit sphere in n-dimensional space is typically denoted S n - 1. Thus, the unit sphere in ordinary 3-space is denoted S2, and the unit circle in the plane is denoted S1. Stone-Weierstrass Theorem   If X is a compact space and C(X) denotes the space of all continuous functions on X, and A is an algebra of functions in C(X) which separates the points of C(X) and which contains a constant function f not identically zero, then A is dense in C(X). open set – Stone-Weierstrass Theorem
 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.