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# Topic: Product topology

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 Direct product - Wikipedia, the free encyclopedia The direct product for vector spaces (not to be confused with the tensor product) is very similar to the one defined for groups above, using the cartesian product with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components. The product topology for infinite products has a twist, and this has to do with being able to make all the projection maps continuous and to make all functions into the product continuous if and only if all its component functions are continuous (i.e. Products (with the product topology) are nice with respect to preserving properties of their factors; for example, the product of Hausdorff spaces is Hausdorff; the product of connected spaces is connected, and the product of compact spaces is compact. en.wikipedia.org /wiki/Direct_product   (955 words)

 Topology -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-06) Topology is concerned with the study of the so-called (Click link for more info and facts about topological properties) topological properties of figures, that is to say properties that do not change under bicontinuous one-to-one transformations (called (Click link for more info and facts about homeomorphism) homeomorphisms). The root of topology was in the study of geometry in ancient cultures. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the "way they are connected together". www.absoluteastronomy.com /encyclopedia/t/to/topology.htm   (1920 words)

 PlanetMath: product topology is not finite, then the product topology is, in general, weaker than the box topology [1]. Another motivation for the product topology comes from category theory: the product topology is simply the direct categorical product of topological spaces. This is version 19 of product topology, born on 2002-06-12, modified 2005-09-29. planetmath.org /encyclopedia/ProductTopology.html   (214 words)

 Weak topology In mathematics, the weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the weakest (that is, smallest) topology on the set which makes all the functions continuous. For example, the product topology is defined to be the weak topology with respect to the projection maps of the product. A particularly important example of a weak topology is that on a normed vector space with respect to its (continuous) dual. www.wordlookup.net /we/weak-topology.html   (424 words)

 Encyclopedia article on Product topology [EncycloZine]   (Site not responding. Last check: 2007-11-06) In topology, the cartesian product of topological spaces is turned into a topological space in the following way. This is easy for finite products, but the statement is (surprisingly) also true for infinite products, when the proof requires the axiom of choice in some form. This shows that the product space is a product in the sense of category theory. encyclozine.com /Product_topology   (763 words)

 [No title] product topology and is equivalent to the axiom of choice. This equivalence shows that the formulation of compactness in infinite product spaces is nonconstructive (also not altogether unexpected, since AC itself is equivalent to asserting whether or not an infinite product is empty!). The right topology, as it turns out, is the cofinite topology with a small twist. en-cyclopedia.com /wiki/Tychonoff's_theorem   (886 words)

 Product Space The topology of p is defined by a base, where a base set in p is given by the cross product of open sets in the component spaces. Once again, verify that the base for the weak topology is valid, and the topology of p includes all finite cross products of open sets in the component spaces, where the remaining components are unconstrained. With the weak topology, the product function is continuous iff the component functions are all continuous. www.mathreference.com /top,prod.html   (1765 words)

 The Box Product Problem   (Site not responding. Last check: 2007-11-06) Given infinitly (countably) many copies of the unit interval, the typical (Tychonov) product topology on their product is topologically a copy of the Hilbert Cube. The topology of the line is generated by open intervals; i.e., (unions of) sets of form (a,b). The box topolgy is defined on all infinite products similarly, and a box product is the product, as sets, of topological spaces given the box topology. www.math.buffalo.edu /~sww/0papers/Box.Product.Problem.html   (161 words)

 [No title]   (Site not responding. Last check: 2007-11-06) This defines > a topology which is at least as fine as the usual product > topology (that is, any set which is open in the usual product > topology is open in this topology). But U is not open in the product topology: every nonempty open set in the product topology contains measurable functions. Subject: Re: sequences in infinite product topology Date: Mon, 15 Mar 1999 12:39:42 -0600 Newsgroups: sci.math On Mon, 15 Mar 1999, Dr. Michael Albert wrote: > Consider the set of all functions from the unit interval > to the reals. www.math.niu.edu /~rusin/known-math/99/product_top   (309 words)

 Topology MAT 530 This is the largest (finest, strongest) topology such that the canonical projection (from the space to the quotient-space) is continuous. The direct (Cartesian) product of two topological spaces also carries a natural direct product topology that is the smallest (coarsest, weakest) topology such that the projections to the factors are continuous. Closed subsets of compact spaces are compact in the subspace topology, the product of two compact spaces is also compact. www.math.sunysb.edu /~timorin/mat530.html   (2904 words)

 Category theory   (Site not responding. Last check: 2007-11-06) Initially, the notions were applied in topology, especially algebraic topology, as an important part of the transition from homology (an intuitive and geometric concept) to homology theory, an axiomatic approach. Tensor products: If C denotes the category of vectorspaces over a fixed field, with linear maps as morphisms, then the tensor product V ⊗ W defines a functor C × C → C which is covariant in both arguments. A category is called cartesian closed if it has finite direct products and a morphism defined on a finite product can always be represented by a morphism defined on just one of the factors. www.sciencedaily.com /encyclopedia/category_theory   (3251 words)

 The product topology is the subspace topology as a subset of R Fortunately this is the same as the topology on the torus thought of as a subset of R A basis for the subspace topology on the torus as a subset of R www-groups.dcs.st-and.ac.uk /~john/MT4522/Lectures/L15.html   (410 words)

 [No title]   (Site not responding. Last check: 2007-11-06) This is difficult but interesting when you take an _arbitrary_ (not finite) product of compact sets, since compactness ought to have something to do with finiteness. Using the box topology the product of compact sets need not be compact). Another classical use of Tychonoff'Theorem in functional analysis is Banach-Alaoglu's Theorem: In the dual space of a Banach space, the unit ball is compact for its weak topology. www.math.niu.edu /~rusin/known-math/95/tychonoff   (488 words)

 The Compact Product Topology   (Site not responding. Last check: 2007-11-06) Usually the product of spaces is taken across some indexing set j. Once the product is established as a set, we apply either the weak or the strong product topology. Let f be a point in the product space p, which is really a function on j, such that f is in the intersection of two base open sets. www.mathreference.com /top-cs,cpt.html   (427 words)

 Product Spaces Diagonal D-branes in product spaces and their Penrose limits... Concentration of measure in product spaces#ÓõãêÝíôñùóç ôïõ ìÝôñïõ óå ÷þñïõò ãéíü... Extension of Isometries in Finite-Dimensional Indefinite Scalar Product Spaces a... www.scienceoxygen.com /math/345.html   (145 words)

 product topology - OneLook Dictionary Search   (Site not responding. Last check: 2007-11-06) We found 3 dictionaries with English definitions that include the word product topology: Tip: Click on the first link on a line below to go directly to a page where "product topology" is defined. Product Topology : Eric Weisstein's World of Mathematics [home, info] www.onelook.com /?w=product+topology   (89 words)

 Topological Equivalents of the Axiom of Choice and of Weak Forms of Choice, by Eric Schechter Tychonov's Product Theorem: any product of compact topological spaces, when equipped with the product topology, is also compact. Some other statements about product topologies are also equivalent: the product of complete uniform spaces, when equipped with the product uniformity, is also complete; the product of closures of subsets of topological spaces is equal to the closure of the product of those subsets. For any set X, the set 2^X (with the product topology) is compact. at.yorku.ca /z/a/a/b/18.htm   (848 words)

 topological vector spaces   (Site not responding. Last check: 2007-11-06) product of a family of topological vector spaces, when endowed with the product topology... S-Neighborhoods and L-Topological Vector Spaces S-Neighborhoods and L-Topological Vector Spaces In this paper the concept of S-neighborhoods is introduced and we give a new characterization of L-topological vector spaces in terms of... Topological Vector Space -- from MathWorld Topological Vector Space -- from MathWorld A vector space with a Hausdorff topology such that the operations of vector addition and scalar multiplication are continuous. www.maps-universe.com /articles/34/topological-vector-spaces.html   (558 words)

 Free Product, Profinite Topology and Finitely Generated Subgroups. (ResearchIndex)   (Site not responding. Last check: 2007-11-06) Free Product, Profinite Topology and Finitely Generated Subgroups. 19 A conjecture on the Hall topology for the free group - Pin, Reuteunauer - 1991 3 Extending partial automorphisms and the profinite topology o.. citeseer.ist.psu.edu /499425.html   (287 words)

 Coreflectively modified continuous duality applied to classical product theorems by Frederic Mynard   (Site not responding. Last check: 2007-11-06) Several (old and new) results on stability of quotients (of various types) under product, on sequentiality of product of sequential spaces, on relationships between a topology and the upper Kuratowski convergence on its closed sets are derived from a general mechanism of duality that uses the continuous convergence. Keywords: Product topology and product maps; Quotient, hereditarily quotient, countably biquotient, biquotient maps; Sequential, strongly sequential, Fr\'echet, strongly Fr\'echet topology; Convergence, product convergence, Antoine convergence, continuous convergence; Cartesian-closedness, exponential objects Copyright © 2001 Applied General Topology and Topology Atlas. at.yorku.ca /i/a/a/j/25.htm   (151 words)

 Outline: Topology M.S. Comprehensive Exam   (Site not responding. Last check: 2007-11-06) Product of two path connected spaces, product of two connected spaces Continuous image of a compact space, the product of two compact spaces, uniform continuity and compactness Topology: A First Course, James R. Munkres, Prentice-Hall. www.math.montana.edu /Documents/Comps/ms_math/Outlines/MS_Topology   (156 words)

 Introduction to Topology — Compare Product Prices & Reviews   (Site not responding. Last check: 2007-11-06) Please note that titles occasionally go out of print or publishers run out of stock. This book is divided into two parts: an introduction to topology, using metric spaces to motivate the definition of a topological space, and the algebraic applications of topology (such as homotopy theory and Jordan's theorem). Another excellent thing is the large number of exercises, most of which have full solutions (!) and/or hints at the end of the book. www.onlinereviewers.co.uk /store/asinsearch_0486406806.htm   (150 words)

 Counterexamples in Topology — Compare Product Prices & Reviews   (Site not responding. Last check: 2007-11-06) In the first part of the book, the authors give a crash course on basic point-set topology. In the second part, the book provides "Counterexamples": quite a few topologies, both the predictable and the... Read reviews and compare prices for Counterexamples in Topology and save www.onlinereviewers.co.uk /store/asinsearch_048668735X.htm   (158 words)

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