A Möbius strip, an object with only one surface and one edge. Such shapes are an object of study in topology.

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.

A three-dimensional depiction of a thickened trefoil knot, the simplest non-trivial knot

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.

History[link]

The Seven Bridges of Königsberg is a famous problem solved by Euler.

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.

A continuous deformation (a type of homeomorphism) of a mug into a doughnut (torus) and back.

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]

Let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:

1. Both the empty set and X are elements of τ
2. Any union of elements of τ is an element of τ
3. 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 X_τ 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]

Topology portal

• 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]

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 Topology

• 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.

Frank Wilczek
Born	(1951-05-15) May 15, 1951 (age 61) Mineola, New York, U.S.
Nationality	United States
Fields	Physics
Institutions	MIT
Alma mater	University of Chicago (B.S.), Princeton University (M.A., Ph.D.)
Doctoral advisor	David Gross
Doctoral students	Mark Alford (*) Michael Forbes Martin Greiter Christoph Holzhey David Kessler Finn Larsen Richard MacKenzie John March-Russell (*) Chetan Nayak Maulik Parikh Krishna Rajagopal David Robertson Sean Robinson Alfred Shapere Stephen Wandzura (*): Jointly a Sidney Coleman student
Known for	Quantum chromodynamics
Notable awards	Lorentz Medal (2002) Nobel Prize in Physics (2004)
Spouse	Betsy Devine

Frank Anthony Wilczek (born May 15, 1951) is a theoretical physicist from the United States and a Nobel laureate. He is currently the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology (MIT).

Wilczek, along with David Gross and H. David Politzer, was awarded the Nobel Prize in Physics in 2004 for their discovery of asymptotic freedom in the theory of the strong interaction.

Biography[link]

Born in Mineola, New York, of Polish and Italian origin, Wilczek was educated in the public schools of Queens, attending Martin Van Buren High School. It was around this time Wilczek's parents realized that he was exceptional - in part as a result of Frank Wilczek having been administered an IQ test. ^[1]

He received his Bachelor of Science in Mathematics at the University of Chicago in 1970, a Master of Arts in Mathematics at Princeton University, 1972, and a Ph.D. in physics at Princeton University in 1974. Wilczek holds the Herman Feshbach Professorship of Physics at MIT Center for Theoretical Physics. He worked at the Institute for Advanced Study in Princeton and the Institute for Theoretical Physics at the University of California, Santa Barbara.

He was awarded the Lorentz Medal in 2002. Wilczek won the Lilienfeld Prize of the American Physical Society in 2003. In the same year he was awarded the Faculty of Mathematics and Physics Commemorative Medal from Charles University in Prague. He was the co-recipient of the 2003 High Energy and Particle Physics Prize of the European Physical Society. Wilczek was also the co-recipient of the 2005 King Faisal International Prize for Science.

He currently serves on the board for Society for Science & the Public.

Wilczek was married to Betsy Devine on July 3, 1973, and together have two daughters, Amity and Mira.

Wilczek has also appeared on an episode of Penn & Teller: Bullshit!, where Penn referred to him as "the smartest person [they have] ever had on the show."

Research[link]

In 1973 Wilczek, a graduate student working with David Gross at Princeton University, discovered asymptotic freedom, which holds that the closer quarks are to each other, the less the strong interaction (or color charge) between them; when quarks are in extreme proximity, the nuclear force between them is so weak that they behave almost as free particles. The theory, which was independently discovered by H. David Politzer, was important for the development of quantum chromodynamics.

Wilczek has helped to reveal and develop axions, anyons, asymptotic freedom, the color superconducting phases of quark matter, and other aspects of quantum field theory. He has worked on an unusually wide range of topics, ranging across condensed matter physics, astrophysics, and particle physics.

Current research

• "Pure" particle physics: connections between theoretical ideas and observable phenomena;
• Behavior of matter: phase structure of quark matter at ultra-high temperature and density; color superconductivity;
• Application of particle physics to cosmology;
• Application of field theory techniques to condensed matter physics;
• Quantum theory of black holes.

Publications[link]

For lay readers[link]

• 2008. The Lightness of Being: Mass, Ether, and the Unification of Forces. Basic Books. ISBN 978-0-465-00321-1.
• 2007. La musica del vuoto. Roma: Di Renzo Editore.
• 2006. Fantastic Realities: 49 Mind Journeys And a Trip to Stockholm. World Scientific. ISBN 978-981-256-655-3.
• 2002, "On the world's numerical recipe (an ode to physics)," Daedalus 131(1): 142-47.
• 1989 (with Betsy Devine). Longing for the Harmonies: Themes and Variations from Modern Physics. W W Norton. ISBN 978-0-393-30596-8.

Technical[link]

• 1988. Geometric Phases in Physics.
• 1990. Fractional Statistics and Anyon Superconductivity.

See also[link]

Wikimedia Commons has media related to: Frank Wilczek

Wikiquote has a collection of quotations related to: Frank Wilczek

• coupling unification
• Quantum chromodynamics
• black holes
• axion
• dark matter
• WIMP
• quantum number
• soliton
• fractional statistics
• Hall effect
• MIT Physics Department
• List of Poles

References[link]

• "Frank Wilczek, Herman Feshbach Professor of Physics". Department of Physics, MIT. 2011. http://web.mit.edu/physics/people/faculty/wilczek_frank.html. Retrieved 2011-06-14. 
• Tore Frängsmyr, editor (2005). location=Stockholm "The Nobel Prizes 2004". Les Prix Nobel. Nobel Foundation. http://nobelprize.org/physics/laureates/2004/wilczek-autobio.html location=Stockholm. Retrieved 2008-04-30. 

External links[link]

• Biography and Bibliographic Resources, from the Office of Scientific and Technical Information, United States Department of Energy
• Longer biography at Lifeboat Foundation website
• Frank Wilczek explains Einstein's massive contributions to science
• Papers in ArXiv
• Frank Wilczek discusses his book "The Lightness of Being" on the 7th Avenue Project Radio Show
• The World's Numerical Recipe
• Scientific articles by Wilczek in the SLAC database
• Wilczek on anyons and superconductivity
• Blog of the Wilczek family's Nobel adventures
• Freeman Dyson, "Leaping into the Grand Unknown: Review of The Lightness of Being," The New York Review of Books 56(6), April 9, 2009.
• ForaTV: The Large Hadron Collider and Unified Field Theory
• A radio interview with Frank Wilczeck Aired on the Lewis Burke Frumkes Radio Show the 10th of April 2011.
• A television interview with Frank Wilczek from February 2011 for Cambridge University Television