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Kaliningrad
Kaliningrad (Russian: Калинингра́д; Polish: Królewiec; Lithuanian: Karaliaučius; German: Königsberg; Czech: Královec) is a seaport city and the administrative center of Kaliningrad Oblast, the Russian exclave between Poland and Lithuania on the Baltic Sea. The territory, the northern part of the former East Prussia, borders on NATO and EU members Poland and Lithuania, and is geographically separated from the rest of Russia.
Originally named Königsberg in German, the town was founded in 1255 by the order of the Teutonic Knights in honour of the Czech king Přemysl Otakar II and was part of Prussia and then of Germany until 1945, but was largely destroyed during World War II. Its ruins were occupied by the Red Army in 1945 and its German population forced out. It was renamed Kaliningrad in 1946 in honor of Mikhail Kalinin.
According to the preliminary results of the 2010 Census, its population was 431,491—an increase from 430,003 recorded in the 2002 Census. Its ethnic composition is 77.9% Russians, 8.0% Belarusians, 7.3% Ukrainians, 1.9% Lithuanians, 0.6% Germans, and 0.5% Poles.
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Cesare Arzelà
Cesare Arzelà (6 March 1847 – 15 March 1912) was an Italian mathematician who taught at the University of Bologna and is recognized for his contributions in the theory of functions, particularly for his characterization of sequences of continuous functions, generalizing the one given earlier by Giulio Ascoli in the famous Arzelà-Ascoli theorem.
He was a pupil of the Scuola Normale Superiore of Pisa where he graduated in 1869. Arzela came from a poor household; therefore he could not start his study until 1871, when he studied in Pisa under Enrico Betti and Ulisse Dini.
He was working in Florence (from 1875) and in 1878 obtained the Chair of Algebra at the University of Palermo.
After that he became a professor in 1880 at the University of Bologna at the department of analysis. He conducted research in the field of theory of functions. His most famous student was Leonida Tonelli.
In 1889 he generalized the Ascoli theorem to Arzelà–Ascoli theorem, an important theorem in theory of functions.
He was a member of the Accademia Nazionale dei Lincei, and of several other academies.
wn.com/Cesare_Arzelà -
Clifford A. Pickover
Clifford Alan Pickover (born 1957) is an American author, editor, and columnist in the fields of science, mathematics, and science fiction, and is employed at the IBM Thomas J. Watson Research Center in Yorktown, New York.
He received his Ph.D. in 1982 from Yale University's Department of Molecular Biophysics and Biochemistry, where he conducted research on X-ray scattering and protein structure. Pickover graduated first in his class from Franklin and Marshall College, after completing the four-year undergraduate program in three years.
He joined IBM at the Thomas J. Watson Research Center in 1982, as a member of the speech synthesis group and later worked on the design-automation workstations. For much of his career, Pickover has published technical articles in the areas of scientific visualization, computer art, and recreational mathematics. Pickover is still employed at the IBM Thomas J. Watson Research Center, where he is the editor of the IBM Journal of Research and Development.
He is currently an associate editor for the scientific journal Computers and Graphics and is an editorial board member for Odyssey and Leonardo. He is also the Brain-Strain columnist for Odyssey magazine, and, for many years, he was the Brain-Boggler columnist for Discover magazine.
wn.com/Clifford_A._Pickover -
Felix Hausdorff
Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.
Hausdorff studied at the University of Leipzig, obtaining his Ph.D. in 1891. He taught mathematics in Leipzig until 1910, when he became professor of mathematics at the University of Bonn. He was professor at the University of Greifswald from 1913 to 1921. He then returned to Bonn. When the Nazis came to power, Hausdorff, who was Jewish, felt that as a respected university professor he would be spared from persecution. However, his abstract mathematics was denounced as "Jewish", useless, and "un-German"[citation needed] and he lost his position in 1935. Though he could no longer publish in Germany, Hausdorff continued to be an active research mathematician, publishing in the Polish journal Fundamenta Mathematicae. After Kristallnacht in 1938 as persecution of Jews escalated, Hausdorff became more and more isolated. He wrote to George Pólya requesting a research fellowship in the United States, but these efforts came to nothing. Finally, in 1942 when he could no longer avoid being sent to a concentration camp, Hausdorff committed suicide together with his wife, Charlotte Goldschmidt Hausdorff, and sister-in-law, Edith Goldschmidt Pappenheim, on 26 January. They are buried in Bonn, Germany.
wn.com/Felix_Hausdorff -
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor (
/ˈkæntɔr/ KAN-tor; German: [ɡeˈɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantɔʁ]; March 3 [O.S. February 19] 1845 – January 6, 1918) was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.
wn.com/Georg_Cantor -
Giulio Ascoli
Giulio Ascoli (20 January 1843, Trieste – 12 July 1896, Milan) was an Italian mathematician. He was a student of the Scuola Normale di Pisa, where he graduated in 1868.
In 1872 he became Professor of Algebra and Calculus of the Politecnico di Milano University. From 1879 he was professor of mathematics at the Reale Istituto Tecnico Superiore, where, in 1901, was affixed a plaque that remembers him.
He was also corresponding member of Istituto Lombardo.
He made contributions to the theory of functions of a real variable and to Fourier series. For example, Ascoli introduced equicontinuity in 1884, a topic regarded as one of the fundamental concepts in the theory of real functions. In 1889, Italian mathematician Cesare Arzelà generalized Ascoli's Theorem into the Arzelà–Ascoli theorem, a practical sequential compactness criterion of functions.
wn.com/Giulio_Ascoli -
Hassler Whitney
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.
Hassler Whitney was the son of First District New York Supreme Court judge Edward Baldwin Whitney and Josepha (Newcomb) Whitney, the grandson of Yale University Professor of Ancient Languages William Dwight Whitney, the great-grandson of Connecticut Governor and US Senator Roger Sherman Baldwin, and the great-great-great-grandson of American founding father Roger Sherman.
Hassler Whitney's maternal grandparents were astronomer and celestial mechanician Simon Newcomb and Mary Hassler Newcomb (the granddaughter of the first superintendent of the Coast Survey Ferdinand Hassler).
He married Margaret R. Howell, May 30, 1930; children: James Newcomb, Carol, Marian; married Mary Barnett Garfield, January 16, 1955; children: Sarah Newcomb, Emily Baldwin; and married Barbara Floyd Osterman, February 8, 1986.
wn.com/Hassler_Whitney -
Henri Poincaré
Jules Henri Poincaré (29 April 1854 – 17 July 1912) (French pronunciation: [ˈʒyl ɑ̃ˈʁi pwɛ̃nkaˈʁe]) was a French mathematician, theoretical physicist, engineer, and a philosopher of science. He is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime.
As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture, which was one of the most famous unsolved problems in mathematics until it was solved in 2002–2003. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology.
Poincaré made clear importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Dutch physicist Hendrik Lorentz (1853–1928) in 1905. Thus he obtained perfect invariance of all of Maxwell's equations, an important step in the formulation of the theory of special relativity.
wn.com/Henri_Poincaré -
Jacques Hadamard
Jacques Salomon Hadamard FRS (8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.
The son of a teacher, Amédée Hadamard, of Jewish descent, and Claire Marie Jeanne Picard, Hadamard was born in Versailles, France and attended the Lycée Charlemagne and Lycée Louis-le-Grand, where his father taught. In 1884 Hadamard entered the École Normale Supérieure, having been placed first in the entrance examinations both there and at the École Polytechnique. His teachers included Tannery, Hermite, Darboux, Appell, Goursat and Picard. He obtained his doctorate in 1892 and in the same year was awarded the Grand Prix des Sciences Mathématiques for his essay on the Riemann zeta function.
In 1892 Hadamard married Louise-Anna Trénel, also of Jewish descent, with whom he had three sons and two daughters. The following year he took up a lectureship in the University of Bordeaux, where he proved his celebrated inequality on determinants, which led to the discovery of Hadamard matrices when equality holds. In 1896 he made two important contributions: he proved the prime number theorem, using complex function theory (also proved independently by Charles Jean de la Vallée-Poussin); and he was awarded the Bordin Prize of the French Academy of Sciences for his work on geodesics in the differential geometry of surfaces and dynamical systems. In the same year he was appointed Professor of Astronomy and Rational Mechanics in Bordeaux. His foundational work on geometry and symbolic dynamics continued in 1898 with the study of geodesics on surfaces of negative curvature. For his cumulative work, he was awarded the Prix Poncelet in 1898.
wn.com/Jacques_Hadamard -
James Munkres
James Raymond Munkres (b. August 18, 1930) is a Professor Emeritus of mathematics at MIT and the author of several texts in the area of topology, including Topology (a well-known undergraduate-level text), Analysis on Manifolds, Elements of Algebraic Topology, and Elementary Differential Topology. He is also the author of Elementary Linear Algebra.
He completed his undergraduate education at Nebraska Wesleyan University and received his Ph.D. from the University of Michigan in 1956; his advisor was Edwin E. Moise. Earlier in his career he taught at the University of Michigan and at Princeton University.
Among his contributions to mathematics is the development of what is sometimes called the Munkres assignment algorithm.
wn.com/James_Munkres -
John L. Kelley
John Leroy Kelley (December 6, 1916, Kansas – November 26, 1999, Oakland, California) was an American mathematician at University of California, Berkeley who worked in general topology and functional analysis.
Kelley's 1955 text, General Topology, which eventually appeared in three editions and several translations, is a classic and widely cited graduate level introduction to topology. An appendix sets out a new approach to axiomatic set theory, now called Morse–Kelley set theory, that builds on Von Neumann–Bernays–Gödel set theory.
After earning B.A. (1936) and M.A. (1937) degrees from the University of California, Los Angeles, he went to the University of Virginia, where he obtained his Ph.D. in 1940. Gordon Whyburn, a student of Robert Lee Moore, supervised his thesis, titled A study of hyperspaces. He taught at the University of Notre Dame until the outbreak of World War II. From 1942 to 1945, he did mathematics for the war effort at the Aberdeen Proving Grounds, where his work unit included his future Berkeley colleagues Anthony Morse and Charles Morrey. After teaching at the University of Chicago, 1946-47, Kelley spent the rest of his career at Berkeley, from which he retired in 1985. He chaired the Mathematics Department at Berkeley 1957-60 and 1975-80. He held visiting appointments at Cambridge University and the Indian Institute of Technology in Kanpur, India.
wn.com/John_L._Kelley -
Kazimierz Kuratowski
Kazimierz Kuratowski (February 2, 1896 – June 18, 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics.
Kazimierz Kuratowski was born in Warsaw, then part of the Russian Empire, on February 2, 1896. He was a son of Marek Kuratow, a barrister, and Róża Karzewski. He completed a Warsaw secondary school, which was named after general Paweł Chrzanowski. In 1913, he enrolled in an engineering course at the University of Glasgow in Scotland, in part because he did not wish to study in Russian; instruction in Polish was prohibited. He completed only one year of study when the outbreak of World War I precluded any further enrollment. In 1915, Russian forces withdrew from Warsaw and Warsaw University was reopened with Polish as the language of instruction. Kuratowski restarted his university education there the same year, this time in mathematics. He obtained his Ph.D. in 1921, in newly independent Poland.
In autumn 1921 Kuratowski was awarded the Ph.D. degree for his groundbreaking work. His thesis statement consisted of two parts. One was devoted to an axiomatic construction of topology via the closure axioms (Sur la notion de l'ensemble fini, "Fundamenta Mathematicae", 1/1920). This first part (republished in a slightly modified form in 1922) has been cited in hundreds of scientific articles.
wn.com/Kazimierz_Kuratowski -
Leonhard Euler
Leonhard Euler (German pronunciation: [ˈɔʏlɐ], ( Swiss German pronunciation (help·info)) ( Standard German pronunciation (help·info)) English approximation, "Oiler"; 15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy.
Euler spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all."
wn.com/Leonhard_Euler -
Vito Volterra
Vito Volterra (3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations.
Born in Ancona, then part of the Papal States, into a very poor Jewish family, Volterra showed early promise in mathematics before attending the University of Pisa, where he fell under the influence of Enrico Betti, and where he became professor of rational mechanics in 1883. He immediately started work developing his theory of functionals which led to his interest and later contributions in integral and integro-differential equations. His work is summarised in his book Theory of functionals and of Integral and Integro-Differential Equations (1930).
In 1892, he became professor of mechanics at the University of Turin and then, in 1900, professor of mathematical physics at the University of Rome La Sapienza. Volterra had grown up during the final stages of the Risorgimento when the Papal States were finally annexed by Italy and, like his mentor Betti, he was an enthusiastic patriot, being named by the king Victor Emmanuel III as a senator of the Kingdom of Italy in 1905. In the same year, he began to develop the theory of dislocations in crystals that was later to become important in the understanding of the behaviour of ductile materials. On the outbreak of World War I, already well into his 50s, he joined the Italian Army and worked on the development of airships under Giulio Douhet. He originated the idea of using inert helium rather than flammable hydrogen and made use of his leadership abilities in organising its manufacture.
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AlgTop0: Introduction to Algebraic Topology
- Order: Reorder
- Duration: 30:05
- Updated: 16 Dec 2012
This is the Introductory lecture to a beginner's course in Algebraic Topology given by NJ Wildberger of the School of Mathematics and Statistics at UNSW in 2010. This first lecture introduces some of the topics of the course and three problems. His YouTube site "Insights into Mathematics" at www.youtube.com under user: njwildberger also contains series on MathFoundations, History of Mathematics, LinearAlgebra, Rational Trigonometry and even one called Elementary Mathematics (K-6) Explained.
http://wn.com/AlgTop0_Introduction_to_Algebraic_Topology
This is the Introductory lecture to a beginner's course in Algebraic Topology given by NJ Wildberger of the School of Mathematics and Statistics at UNSW in 2010. This first lecture introduces some of the topics of the course and three problems. His YouTube site "Insights into Mathematics" at www.youtube.com under user: njwildberger also contains series on MathFoundations, History of Mathematics, LinearAlgebra, Rational Trigonometry and even one called Elementary Mathematics (K-6) Explained.
Topology - Part 1
- Order: Reorder
- Duration: 14:58
- Updated: 17 Dec 2012
Part 1 of the Topology series from Why U. A humorous look at the topology of curved space. For more information visit www.WhyU.org
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Part 1 of the Topology series from Why U. A humorous look at the topology of curved space. For more information visit www.WhyU.org
- published: 05 Jul 2011
- views: 45860
- author: MyWhyU
Bob Franzosa - Introduction to Topology
- Order: Reorder
- Duration: 54:04
- Updated: 13 Dec 2012
www.coa.edu 2010.02.09 Introduction to Topology From the Konigsberg Bridges to Geographic Information Systems. Topology is the study of the spatial properties of objects that are unchanged by stretching. It is considered one of the essential fields of modern pure mathematics, and has found many applications outside of mathematics as well.
http://wn.com/Bob_Franzosa_Introduction_to_Topology
www.coa.edu 2010.02.09 Introduction to Topology From the Konigsberg Bridges to Geographic Information Systems. Topology is the study of the spatial properties of objects that are unchanged by stretching. It is considered one of the essential fields of modern pure mathematics, and has found many applications outside of mathematics as well.
- published: 12 Feb 2010
- views: 21337
- author: collegeoftheatlantic
Topology: Jordan's Curve Theorem
- Order: Reorder
- Duration: 1:58
- Updated: 12 Dec 2012
Excerpt from the IBM film "Mathematics Peepshow".
http://wn.com/Topology_Jordan's_Curve_Theorem
Excerpt from the IBM film "Mathematics Peepshow".
- published: 12 Jan 2011
- views: 5939
- author: jbmathfunnew
General Topology Lecture 1 Part 1
- Order: Reorder
- Duration: 13:40
- Updated: 17 Dec 2012
Lecture written by Victor Victorov and edited and presented by James Dilts Definition and Examples
http://wn.com/General_Topology_Lecture_1_Part_1
Lecture written by Victor Victorov and edited and presented by James Dilts Definition and Examples
- published: 16 Mar 2011
- views: 20306
- author: vivictorov
Hitler Learns Topology
- Order: Reorder
- Duration: 3:50
- Updated: 17 Dec 2012
Hitler gets confused about the topological definitions of open and closed sets. Then he totally freaks out.
http://wn.com/Hitler_Learns_Topology
Hitler gets confused about the topological definitions of open and closed sets. Then he totally freaks out.
- published: 01 Feb 2010
- views: 91750
- author: Benjatastic
Introduction to Topology
- Order: Reorder
- Duration: 8:30
- Updated: 11 Dec 2012
This first lecture of the series Comically Small Lectures On Point Set Topology. We will give: the definition of a topology, open and closed sets, and the basis for a topology. Examples of topologies, including the trivial topology, the discrete topology, particular point / excluded point topologies.
http://wn.com/Introduction_to_Topology
This first lecture of the series Comically Small Lectures On Point Set Topology. We will give: the definition of a topology, open and closed sets, and the basis for a topology. Examples of topologies, including the trivial topology, the discrete topology, particular point / excluded point topologies.
- published: 13 May 2009
- views: 44576
- author: jadenbane
Computer Networking Tutorial - 3 - Network Topology
- Order: Reorder
- Duration: 6:59
- Updated: 15 Dec 2012
Introduction to computer networking. Any questions about the video? I will answer all questions at thenewboston.com
http://wn.com/Computer_Networking_Tutorial_3_Network_Topology
Introduction to computer networking. Any questions about the video? I will answer all questions at thenewboston.com
- published: 31 Jan 2009
- views: 214079
- author: thenewboston
AlgTop0a: Introduction to Algebraic Topology
- Order: Reorder
- Duration: 9:50
- Updated: 03 Dec 2012
NOTE: This entire series is now available in the full (hour long lectures), also at this channel under the playlist AlgTop (full lectures): www.youtube.com This is the first video of the introductory lecture of a beginner's course in Algebraic Topology. The subject is one of the most dynamic and exciting areas of 20th century mathematics, with its roots in the work of Riemann, Klein and Poincare in the latter half of the 19th century. This first lecture will outline the main topics, and will present three well-known but perhaps challenging problems for you to try. The course is for 3rd or 4th year undergraduate math students, but anyone with some mathematical maturity and a little background or willingness to learn group theory can benefit. The subject is particularly important for modern physics. Our treatment will have many standard features, but also some novelties. The lecturer is Assoc Prof NJ Wildberger of the School of Mathematics and Statistics at UNSW, Sydney, Australia.
http://wn.com/AlgTop0a_Introduction_to_Algebraic_Topology
NOTE: This entire series is now available in the full (hour long lectures), also at this channel under the playlist AlgTop (full lectures): www.youtube.com This is the first video of the introductory lecture of a beginner's course in Algebraic Topology. The subject is one of the most dynamic and exciting areas of 20th century mathematics, with its roots in the work of Riemann, Klein and Poincare in the latter half of the 19th century. This first lecture will outline the main topics, and will present three well-known but perhaps challenging problems for you to try. The course is for 3rd or 4th year undergraduate math students, but anyone with some mathematical maturity and a little background or willingness to learn group theory can benefit. The subject is particularly important for modern physics. Our treatment will have many standard features, but also some novelties. The lecturer is Assoc Prof NJ Wildberger of the School of Mathematics and Statistics at UNSW, Sydney, Australia.
- published: 22 Jul 2010
- views: 10836
- author: njwildberger
Topology #1 Metrics
- Order: Reorder
- Duration: 8:07
- Updated: 17 Dec 2012
Introduction to Metrics
http://wn.com/Topology_1_Metrics
CompTIA Network+ Training: Network Topologies
- Order: Reorder
- Duration: 10:01
- Updated: 09 Dec 2012
This Network+ Training video is from Network+ (N10-004) LearnSmart Video Training. Check out LearnSmart's unlimited access training at: www.learnsmartsystems.com For more Network+ Certification training videos and tutorials.
http://wn.com/CompTIA_Network+_Training_Network_Topologies
This Network+ Training video is from Network+ (N10-004) LearnSmart Video Training. Check out LearnSmart's unlimited access training at: www.learnsmartsystems.com For more Network+ Certification training videos and tutorials.
- published: 16 Nov 2009
- views: 26075
- author: preplogic
Through one hole or two?
- Order: Reorder
- Duration: 0:45
- Updated: 16 Dec 2012
A topological magic trick
http://wn.com/Through_one_hole_or_two?
Kate Poirier: Compactifying String Topology
- Order: Reorder
- Duration: 74:11
- Updated: 25 Jul 2012
Kate Poirier, University of California - Berkeley String topology studies the algebraic topology of the free loop space of a closed, oriented manifold. Previous treatments of string topology describe algebraic structures on the homology of the free loop space of the manifold and operations parameterized by a noncompact space of graphs. One perspective is that these structures should be a shadow of a richer structure at the chain level and that the space parametrizing the operations should be compactied. In this talk, we describe the compact space of graphs giving string topology operations on the singular chains of the free loop space which induce known operations on homology. This is joint work with Nathaniel Rounds.
http://wn.com/Kate_Poirier_Compactifying_String_Topology
Kate Poirier, University of California - Berkeley String topology studies the algebraic topology of the free loop space of a closed, oriented manifold. Previous treatments of string topology describe algebraic structures on the homology of the free loop space of the manifold and operations parameterized by a noncompact space of graphs. One perspective is that these structures should be a shadow of a richer structure at the chain level and that the space parametrizing the operations should be compactied. In this talk, we describe the compact space of graphs giving string topology operations on the singular chains of the free loop space which induce known operations on homology. This is joint work with Nathaniel Rounds.
- published: 25 Jul 2012
- views: 432
- author: NDdotEDU
Common Physical Network Topologies - CompTIA Network+ N10-004: 2.3
- Order: Reorder
- Duration: 9:38
- Updated: 12 Nov 2012
See our entire index of CompTIA Network+ videos at www.FreeNetworkPlus.com - Networking fundamentals include many different physical network topology types. In this video, we'll detail the basic structure of star networks, mesh networks, ring networks, and much more.
http://wn.com/Common_Physical_Network_Topologies_CompTIA_Network+_N10-004_23
See our entire index of CompTIA Network+ videos at www.FreeNetworkPlus.com - Networking fundamentals include many different physical network topology types. In this video, we'll detail the basic structure of star networks, mesh networks, ring networks, and much more.
- published: 18 Jul 2010
- views: 17945
- author: professormesser
Network topology
- Order: Reorder
- Duration: 0:26
- Updated: 24 Nov 2012
logical topology
http://wn.com/Network_topology
MathHistory17: Topology
- Order: Reorder
- Duration: 55:48
- Updated: 16 Dec 2012
This video is the last in this series on the History of Mathematics, and gives a brief introduction to Topology. The subject goes back to Euler (as do so many things in modern mathematics) with his discovery of the Euler characteristic of a polyhedron, although arguably Descartes had found something close to this in his analysis of curvature of a polyhedron. We introduce this via rational turn angles, a renormalization of angle where a full turn has the value one (very reasonable, and ought to be used more!!) The topological nature of the Euler characteristic was perhaps first understood by Poincare, and we sketch his argument for its invariance under continuous transformations. We discuss the sphere, torus, genus g surfaces and the classification of orientable, and non-orientable closed 2 dimensional surfaces, such as the Mobius band (which has a boundary) and the projective plane (which does not). The interest in these objects resulted from Riemann's work on surfaces associated to multi-valued functions in the setting of complex analysis. Finally we briefly mention the important notion of a simply connected space, and the Poincare conjecture, solved recently by G. Perelman. If you enjoy this subject, you can have a look at my video series Algebraic Topology. Thanks for watching the series!
http://wn.com/MathHistory17_Topology
This video is the last in this series on the History of Mathematics, and gives a brief introduction to Topology. The subject goes back to Euler (as do so many things in modern mathematics) with his discovery of the Euler characteristic of a polyhedron, although arguably Descartes had found something close to this in his analysis of curvature of a polyhedron. We introduce this via rational turn angles, a renormalization of angle where a full turn has the value one (very reasonable, and ought to be used more!!) The topological nature of the Euler characteristic was perhaps first understood by Poincare, and we sketch his argument for its invariance under continuous transformations. We discuss the sphere, torus, genus g surfaces and the classification of orientable, and non-orientable closed 2 dimensional surfaces, such as the Mobius band (which has a boundary) and the projective plane (which does not). The interest in these objects resulted from Riemann's work on surfaces associated to multi-valued functions in the setting of complex analysis. Finally we briefly mention the important notion of a simply connected space, and the Poincare conjecture, solved recently by G. Perelman. If you enjoy this subject, you can have a look at my video series Algebraic Topology. Thanks for watching the series!
- published: 13 May 2012
- views: 3995
- author: njwildberger
Topology #3 Metric Examples (Part 2)
- Order: Reorder
- Duration: 11:51
- Updated: 17 Nov 2012
The Discrete Metric, the Max Metric and the Taxicab Metric.
http://wn.com/Topology_3_Metric_Examples_Part_2
The Discrete Metric, the Max Metric and the Taxicab Metric.
- published: 26 Oct 2010
- views: 4763
- author: ThoughtSpaceZero
Blender Dynamic Topology Sculpting
- Order: Reorder
- Duration: 0:50
- Updated: 13 Dec 2012
Quick demonstration of adaptive topology sculpting based off the paper "Freestyle: Sculpting meshes with self-adaptive topology" by Lucian Stãnculescua, Raphaëlle Chainea, and Marie-Paule Canic. www.sciencedirect.com The code is available from Gitorious (in the adaptive branch): gitorious.org See also this post: code.blender.org
http://wn.com/Blender_Dynamic_Topology_Sculpting
Quick demonstration of adaptive topology sculpting based off the paper "Freestyle: Sculpting meshes with self-adaptive topology" by Lucian Stãnculescua, Raphaëlle Chainea, and Marie-Paule Canic. www.sciencedirect.com The code is available from Gitorious (in the adaptive branch): gitorious.org See also this post: code.blender.org
- published: 17 Oct 2011
- views: 17049
- author: BlenderCoders
LTE Network Topology
- Order: Reorder
- Duration: 5:38
- Updated: 10 Dec 2012
LTE Network Topology
http://wn.com/LTE_Network_Topology
Topology - Part 2
- Order: Reorder
- Duration: 8:18
- Updated: 15 Dec 2012
Part 2 of the Topology series from Why U. A humorous look at the topology of curved space. For more information visit www.WhyU.org
http://wn.com/Topology_Part_2
Part 2 of the Topology series from Why U. A humorous look at the topology of curved space. For more information visit www.WhyU.org
- published: 05 Jul 2011
- views: 13221
- author: MyWhyU
Advanced Mobius Strip Topology (19087512 -210)
- Order: Reorder
- Duration: 9:09
- Updated: 01 Dec 2012
I've cut a 3 flip mobius counter clockwise strip down the middle using special mobius paper that I created. I then added half cuts to the plain so they can intersect for the purpose of getting rid of all twists. As you know, the resulting loop has 8 twists, a trefoil knot and two boundaries (surfaces), but it takes 6 lines to kill the 8 twists. The resulting paper soultion is then put on a page and turned into a knot diagram and knot notation. The knot diagram is then turned into a pseudograph consisting of vertices, lines (I meant to make them straight but I do this in my head) and loops. The findings are very very interesting! All the solutions have a knot crossing number of six (except for one so far - it had 8?), all pseudographs have a network Betti number of 7 and almost all the pseudographs have a Euler characteristic of 0. The original uncut mobius strip has a Euler number of 0! WOW. I think all of these findings are reproduceable and original because I cannot find any related papers so far. Can anyone help?
http://wn.com/Advanced_Mobius_Strip_Topology_19087512_-210
I've cut a 3 flip mobius counter clockwise strip down the middle using special mobius paper that I created. I then added half cuts to the plain so they can intersect for the purpose of getting rid of all twists. As you know, the resulting loop has 8 twists, a trefoil knot and two boundaries (surfaces), but it takes 6 lines to kill the 8 twists. The resulting paper soultion is then put on a page and turned into a knot diagram and knot notation. The knot diagram is then turned into a pseudograph consisting of vertices, lines (I meant to make them straight but I do this in my head) and loops. The findings are very very interesting! All the solutions have a knot crossing number of six (except for one so far - it had 8?), all pseudographs have a network Betti number of 7 and almost all the pseudographs have a Euler characteristic of 0. The original uncut mobius strip has a Euler number of 0! WOW. I think all of these findings are reproduceable and original because I cannot find any related papers so far. Can anyone help?
- published: 04 Jun 2008
- views: 9285
- author: David Sparks
1/7 - Facial Intro Topology - Sintel
- Order: Reorder
- Duration: 6:45
- Updated: 13 Dec 2012
Part 1 of 7, created by Angela Guenette for "Sintel" DVD making-of extras, released September 2010. About this project: www.sintel.org Watch "Sintel" on YouTube youtu.be 4-disk DVD box : www.blender3d.org
http://wn.com/1_7_Facial_Intro_Topology_Sintel
Part 1 of 7, created by Angela Guenette for "Sintel" DVD making-of extras, released September 2010. About this project: www.sintel.org Watch "Sintel" on YouTube youtu.be 4-disk DVD box : www.blender3d.org
- published: 17 Jun 2011
- views: 33124
- author: PonderStudios
star topology
- Order: Reorder
- Duration: 0:23
- Updated: 30 Nov 2012
logical
http://wn.com/star_topology
photo: AP / Jason DeCrow
16 Dec 2012
NEWTOWN, Conn. — The gunman in the Connecticut shooting rampage committed suicide as first responders closed in, the governor said Sunday, raising the specter that Adam Lanza had planned an even more gruesome massacre and was stopped short. Lanza blasted his way into the building and used a high-power rifle to kill 20 children and six adults, including the principal who tried to stop him, authorities said ... Among the questions ... H ... The Rev....(size: 8.8Kb)
photo: AP / Guang Niu, Pool
16 Dec 2012
Retired Col. Gen. Viktor Yesin yesterday wrapped up a visit to the United States during which he met with officials at the Pentagon, State Department as well as members of Congress warning about underestimating China’s tactical nuclear weapons buildup ... Yesin warns that U.S. and Western arms specialists are dramatically underestimating China’s nuclear arsenal ... Somewhere between the higher and lower estimates is no doubt the truth ... Mr ... Mr ... ....(size: 4.0Kb)
photo: AP / David Goldman
17 Dec 2012
WASHINGTON—Democrats say meaningful action in the wake of last week's elementary school shooting must include a ban on military-style assault weapons and a look at how the nation deals with individuals suffering from serious mental illness. Several Democratic lawmakers and Independent Sen ... Gun control was a hot topic in the early 1990s, when Congress enacted a 10-year ban on assault weapons ... He added ... Sen ... Rep ... ———....(size: 5.3Kb)
photo: AP / Newtown Bee, Shannon Hicks
15 Dec 2012
Newtown, Conn. Twenty schoolchildren were slaughtered by a heavily armed gunman who opened fire at a suburban elementary school in Connecticut on Friday, ultimately killing at least 27 people including himself in the one of the worst mass shootings in US history ... Authorities found 18 children and seven adults, including the gunman, dead at the school, and two children were pronounced dead later after being taken to a hospital ... Reuters ....(size: 7.5Kb)
photo: AP / Jessica Hill
14 Dec 2012
More than two dozen people, mostly elementary school children, were shot and killed at a Newtown, Conn., elementary school this morning, federal and state sources tell ABC News. The massacre involved two gunmen at a Connecticut school this morning, prompting the town of Newtown to lock down all of its schools and draw SWAT teams to the school, authorities said today. State Police confirm that one shooter is dead ... at ABCNews.com ... ....(size: 4.5Kb)
noodls
17 Dec 2012
The APAX-507x series feature an integrated 2-port Ethernet Switch, which supports a flexible topology connection. For example, the I/O nodes can be connected in a line (bus) topology, instead of a traditional star topology, therefore reducing the effort required ......(size: 1.6Kb)
noodls
17 Dec 2012
(Source. Oklahoma State University). Two mathematicians from Oklahoma State University recently earned selection to the initial class of American Mathematical Society Fellows for 2013 in recognition of their international excellence in mathematical science and service ... Friedlander said ... Kerr Professor at OSU ... His interests are in low-dimensional topology, geometry, combinatorial and geometric group theory, and computational complexity....(size: 3.5Kb)
noodls
17 Dec 2012
(Source. National Cheng Kung University). Tainan, Taiwan, Dec. 17, 2012 ... It helps dialysis centers monitor the physiological status of dialysis patients ... Chen ... The reasons for using ZigBee, Chen said, are its low electricity consumption, low cost, its ability to support large amount of network nodes and different network topologies as well as its low complexity, high speed, high reliability, and high safety level ... Enditem/sc. distributed by....(size: 4.7Kb)
Yahoo Daily News
17 Dec 2012
Online question and answer format proves popular for small businesses. Traffic grows 335% in 2012. Boston, MA (PRWEB) December 17, 2012 The JM Internet Group (web. jm-seo.org), a leader in providing SEO Tips for Small Business via innovative trainings, is proud to announce that Google / SEO traffic to its popular SEO tips section has grown over 335% percent in 2012 ... Top Ten ... Keywords ... News ... Creating the Best Topology for Google Rank ... Web ... ....(size: 3.8Kb)
Cray Awarded $39 Million Supercomputer Contract From the North-German Supercomputing Alliance (HLRN)
Stockhouse17 Dec 2012
The Cray XC30 supercomputer features the new HPC-optimized Aries system interconnect; a new Dragonfly topology that frees applications from locality constraints; an innovative cooling system that utilizes a transverse airflow to lower customers' total cost of ownership; the next-generation of the scalable, high performance Cray Linux Environment ......(size: 6.5Kb)
Yahoo Daily News
16 Dec 2012
Exciting YouTube video explaining top SEO tips for Google reaches 2,500 YouTube views. Los Angeles, CA (PRWEB) December 16, 2012 The JM Internet Group (web. jm-seo.org), a leader in providing Google SEO tips for small business marketers, is proud to announce that their 'Top Five SEO Tips Video' has surpassed 2,500 views on YouTube ... To watch this free video on Google SEO tips, go to - ... Top Ten ... Creating the Best Topology for Google Rank ... ....(size: 3.5Kb)
noodls
15 Dec 2012
In recent years, this group has spearheaded many of the experimental breakthroughs in the burgeoning field of topological insulators, positioning them at the forefront of the knowledge frontier. Topological insulators are highly unusual materials featuring interiors that do not conduct electricity and metallic surfaces that are highly conductive, owing to the presence of special electrons that behave in extraordinary ways....(size: 8.9Kb)
San Francisco Chronicle
15 Dec 2012
LiveAction Visual Network Management Tool Recognized for Innovation. Honolulu, HI (PRWEB) December 14, 2012 ... Maps of devices create a visualization of the network, from the intricacies of a single physical device to multi-layer topological information on connections, traffic, and routes ... LiveAction Flow. Offers a network topology view providing a unique end-to-end flow visualization of live traffic across the network ... LiveAction LAN ... ###....(size: 7.0Kb)
PhysOrg
14 Dec 2012
That's basically the width of a chain of atoms, Yakobson said. The discovery reported this week in the American Chemical Society journal Nano Letters sprang from an investigation of how atoms energetically relate to each other and form topological defects in two-dimensional semiconductors ... Credit ... "It is important to understand the effects of topological defects on the electronic properties as we push toward post-silicon devices.". ....(size: 4.2Kb)
noodls
14 Dec 2012
(Source. Rice University) ... Graphic by the Yakobson Group/Rice University ... The discovery reported this week in the American Chemical Society journal Nano Letters sprang from an investigation of how atoms energetically relate to each other and form topological defects in two-dimensional semiconductors ... "It is important to understand the effects of topological defects on the electronic properties as we push toward post-silicon devices."....(size: 6.5Kb)
Business Wire
13 Dec 2012
Cloud Experience Manager Recognized for Innovation ... Its Cloud Experience Manager (CEM) does more than monitoring; it provides complete network information correlation of session, content and IP topology data in real time ... About NetSocket ... CEM is more than just monitoring, it provides complete network information correlation, automatically correlating session, content and IP topology data in real-time without probes ... About TMC. ... or ... 228 ... ....(size: 11.7Kb)
The Examiner
13 Dec 2012
Rating for Evo ported throttle body by GrimmSpeed. 5 ... Official GrimmSpeed product page. Evo ported and ceramic coated throttle body ... Back to the original article ... Every corner is covered, every surface is permeated; despite the vast topology found on the throttle body, it comes out looking absolutely superb! Thermal protection is also very important here as the throttle body is made out of aluminum as well as the intake manifold ... ....(size: 2.5Kb)
noodls
13 Dec 2012
(Source. Supertex Inc) FOR IMMEDIATE RELEASE Corporate Headquarters.Christine Mahiya. Marcom Manager. 408/222-4858 christin@supertex.com ISOLATED LED DRIVER FROM SUPERTEX REDUCES COMPONENT COUNT IN GENERAL SOLID-STATE LIGHTING APPLICATIONS. HV9973 Requires No Secondary Components ... HV9973 controls an isolated, flyback topology that meets UL requirements for galvanic isolation and operates from 280-400VDC ... Samples are available from stock....(size: 3.0Kb)
Not to be confused with topography.
For other uses, see Topology (disambiguation).
Topology (from the Greek τόπος, “place”, and λόγος, “study”) is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation.
Ideas that are now classified as topological were expressed as early as 1736. Toward the end of the 19th century, a distinct discipline developed, which was referred to in Latin as the geometria situs (“geometry of place”) or analysis situs (Greek-Latin for “picking apart of place”). This later acquired the modern name of topology. By the middle of the 20th century, topology had become an important area of study within mathematics.
The word topology is used both for the mathematical discipline and for a family of sets with certain properties that are used to define a topological space, a basic object of topology. Of particular importance are homeomorphisms, which can be defined as continuous functions with a continuous inverse.
Topology includes many subfields. The most basic and traditional division within topology is point-set topology, which establishes the foundational aspects of topology and investigates concepts inherent to topological spaces (basic examples include compactness and connectedness); algebraic topology, which generally tries to measure degrees of connectivity using algebraic constructs such as homotopy groups and homology; and geometric topology, which primarily studies manifolds and their embeddings (placements) in other manifolds. Some of the most active areas, such as low dimensional topology and graph theory, do not fit neatly in this division. Knot theory studies mathematical knots.
See also: topology glossary for definitions of some of the terms used in topology and topological space for a more technical treatment of the subject.
Contents |
History[link]
Topology began with the investigation of certain questions in geometry. Leonhard Euler's 1736 paper on the Seven Bridges of Königsberg[1] is regarded as one of the first academic treatises in modern topology.
The term "Topologie" was introduced in German in 1847 by Johann Benedict Listing in Vorstudien zur Topologie,[2] who had used the word for ten years in correspondence before its first appearance in print. "Topology," its English form, was first used in 1883 in Listing's obituary in the journal Nature[3] to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". The term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator.[citation needed] However, none of these uses corresponds exactly to the modern definition of topology.
Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. Cantor, in addition to establishing the basic ideas of set theory, considered point sets in Euclidean space as part of his study of Fourier series.
Henri Poincaré published Analysis Situs in 1895,[4] introducing the concepts of homotopy and homology, which are now considered part of algebraic topology.
Maurice Fréchet, unifying the work on function spaces of Cantor, Volterra, Arzelà, Hadamard, Ascoli, and others, introduced the metric space in 1906.[5] A metric space is now considered a special case of a general topological space. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space.[6] In current usage, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski.[citation needed]
For further developments, see point-set topology and algebraic topology.
Elementary introduction[link]
Topology, as a branch of mathematics, can be formally defined as "the study of qualitative properties of certain objects (called topological spaces) that are invariant under certain kind of transformations (called continuous maps), especially those properties that are invariant under a certain kind of equivalence (called homeomorphism)." To put it more simply, topology is the study of continuity and connectivity.
The term topology is also used to refer to a structure imposed upon a set X, a structure that essentially 'characterizes' the set X as a topological space by taking proper care of properties such as convergence, connectedness and continuity, upon transformation.
Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of the great unifying ideas of mathematics.
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 put together. For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside.
One of the first papers in topology was the demonstration, by Leonhard Euler, that it was impossible to find a route through the town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges, nor on their distance from one another, but only on connectivity properties: which bridges are connected to which islands or riverbanks. This problem, the Seven Bridges of Königsberg, is now a famous problem in introductory mathematics, and led to the branch of mathematics known as graph theory.
Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick." This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the exact shape of the sphere; it applies to pear shapes and in fact any kind of smooth blob, as long as it has no holes.
To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of homeomorphism. The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and the hairy ball theorem applies to any space homeomorphic to a sphere.
Intuitively two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist can't distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle. A precise definition of homeomorphic, involving a continuous function with a continuous inverse, is necessarily more technical.
Homeomorphism can be considered the most basic topological equivalence. Another is homotopy equivalence. This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from "squishing" some larger object.
| Equivalence classes of the English alphabet: | |
|---|---|
| Homeomorphism | Homotopy equivalence |
 |
 |
An introductory exercise is to classify the uppercase letters of the English alphabet according to homeomorphism and homotopy equivalence. The result depends partially on the font used. The figures use a sans-serif font named Myriad. Notice that homotopy equivalence is a rougher relationship than homeomorphism; a homotopy equivalence class can contain several of the homeomorphism classes. The simple case of homotopy equivalence described above can be used here to show two letters are homotopy equivalent, e.g. O fits inside P and the tail of the P can be squished to the "hole" part.
Thus, the homeomorphism classes are: one hole two tails, two holes no tail, no holes, one hole no tail, no holes three tails, a bar with four tails (the "bar" on the K is almost too short to see), one hole one tail, and no holes four tails.
The homotopy classes are larger, because the tails can be squished down to a point. The homotopy classes are: one hole, two holes, and no holes.
To be sure we have classified the letters correctly, we not only need to show that two letters in the same class are equivalent, but that two letters in different classes are not equivalent. In the case of homeomorphism, this can be done by suitably selecting points and showing their removal disconnects the letters differently. For example, X and Y are not homeomorphic because removing the center point of the X leaves four pieces; whatever point in Y corresponds to this point, its removal can leave at most three pieces. The case of homotopy equivalence is harder and requires a more elaborate argument showing an algebraic invariant, such as the fundamental group, is different on the supposedly differing classes.
Letter topology has some practical relevance in stencil typography. The font Braggadocio, for instance, has stencils that are made of one connected piece of material.
Mathematical definition[link]
Main article: Topological space
Let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:
- Both the empty set and X are elements of τ
- Any union of elements of τ is an element of τ
- Any intersection of finitely many elements of τ is an element of τ
If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation Xτ may be used to denote a set X endowed with the particular topology τ.
The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (i.e., its complement is open). A subset of X may be open, closed, both (clopen set), or neither. The empty set and X itself are always clopen.
A function or map from one topological space to another is called continuous if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both spaces with the Standard Topology), then this definition of continuous is equivalent to the definition of continuous in calculus. If a continuous function is one-to-one and onto, and if the inverse of the function is also continuous, then the function is called a homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties, and are considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. But the circle is not homeomorphic to the doughnut.
Topology topics[link]
Some theorems in general topology[link]
- Every closed interval in R of finite length is compact. More is true: In Rn, a set is compact if and only if it is closed and bounded. (See Heine–Borel theorem).
- Every continuous image of a compact space is compact.
- Tychonoff's theorem: the (arbitrary) product of compact spaces is compact.
- A compact subspace of a Hausdorff space is closed.
- Every continuous bijection from a compact space to a Hausdorff space is necessarily a homeomorphism.
- Every sequence of points in a compact metric space has a convergent subsequence.
- Every interval in R is connected.
- Every compact finite-dimensional manifold can be embedded in some Euclidean space Rn.
- The continuous image of a connected space is connected.
- Every metric space is paracompact and Hausdorff, and thus normal.
- The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.
- The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.
- Any open subspace of a Baire space is itself a Baire space.
- The Baire category theorem: If X is a complete metric space or a locally compact Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty.
- On a paracompact Hausdorff space every open cover admits a partition of unity subordinate to the cover.
- Every path-connected, locally path-connected and semi-locally simply connected space has a universal cover.
General topology also has some surprising connections to other areas of mathematics. For example:
- In number theory, Fürstenberg's proof of the infinitude of primes.
See also some counter-intuitive theorems, e.g. the Banach–Tarski one.
Some useful notions from algebraic topology[link]
See also list of algebraic topology topics.
- Homology and cohomology: Betti numbers, Euler characteristic, degree of a continuous mapping.
- Operations: cup product, Massey product
- Intuitively attractive applications: Brouwer fixed-point theorem, Hairy ball theorem, Borsuk–Ulam theorem, Ham sandwich theorem.
- Homotopy groups (including the fundamental group).
- Chern classes, Stiefel–Whitney classes, Pontryagin classes.
Generalizations[link]
Occasionally, one needs to use the tools of topology but a "set of points" is not available. In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are certain structures defined on arbitrary categories that allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.
See also[link]
- List of algebraic topology topics
- List of general topology topics
- List of geometric topology topics
- List of topology topics
- Publications in topology
- Topology glossary
- General topology
- List of examples in general topology
References[link]
- ^ Euler, Leonhard, Solutio problematis ad geometriam situs pertinentis
- ^ Listing, Johann Benedict, "Vorstudien zur Topologie", Vandenhoeck und Ruprecht, Göttingen, p. 67, 1848
- ^ Tait, Peter Guthrie, "Johann Benedict Listing (obituary)", Nature *27*, 1 February 1883, pp. 316–317
- ^ Poincaré, Henri, "Analysis situs", Journal de l'École Polytechnique ser 2, 1 (1895) pp. 1–123
- ^ Fréchet, Maurice, "Sur quelques points du calcul fonctionnel", PhD dissertation, 1906
- ^ Hausdorff, Felix, "Grundzüge der Mengenlehre", Leipzig: Veit. In (Hausdorff Werke, II (2002), 91–576)
Further reading[link]
- Basener, William (2006). Topology and Its Applications (1st ed.). Wiley. ISBN 0-471-68755-3.
- Bourbaki; Elements of Mathematics: General Topology, Addison–Wesley (1966).
- Breitenberger, E. (2006). "Johann Benedict Listing". In I.M. James. History of Topology. North Holland. ISBN 978-0-444-82375-5.
- Kelley, John L. (1975). General Topology. Springer-Verlag. ISBN 0-387-90125-6.
- Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.
- Pickover, Clifford A. (2006). The Möbius Strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology. Thunder's Mouth Press (Provides a popular introduction to topology and geometry). ISBN 1-56025-826-8.
- Boto von Querenburg (2006). Mengentheoretische Topologie. Heidelberg: Springer-Lehrbuch. ISBN 3-540-67790-9 (German)
- Richeson, David S. (2008) Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press.
- Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.
External links[link]
 |
Wikimedia Commons has media related to: Topology |
 |
Wikibooks has more on the topic of |
- Elementary Topology: A First Course Viro, Ivanov, Netsvetaev, Kharlamov
- Topology at the Open Directory Project
- The Topological Zoo at The Geometry Center
- Topology Atlas
- Topology Course Lecture Notes Aisling McCluskey and Brian McMaster, Topology Atlas
- Topology Glossary
- Moscow 1935: Topology moving towards America, a historical essay by Hassler Whitney.
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This article is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License, which means that you can copy and modify it as long as the entire work (including additions) remains under this license.
