A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. This article is about the mathematical object See Mobius Band (music group for the music group

Topology (Greek topos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure. Topology builds on set theory, considering both sets of points and families of sets.

The word topology is used both for the area of study and for a family of sets with certain properties described below that are used to define a topological space. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Of particular importance in the study of topology are functions or maps that are homeomorphisms. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Topological equivalence redirects here see also Topological equivalence (dynamical systems. Informally, these functions can be thought of as those that stretch space without tearing it apart or sticking distinct parts together.

When the discipline was first properly founded, toward the end of the 19th century, it was called geometria situs (Latin geometry of place) and analysis situs (Latin analysis of place). The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. From around 1925 to 1975 it was an important growth area within mathematics.

Topology is a large branch of mathematics that includes many subfields. The most basic division within topology is point-set topology, which investigates such concepts as compactness, connectedness, and countability; algebraic topology, which investigates such concepts as homotopy and homology; and geometric topology, which studies manifolds and their embeddings, including knot theory. In Mathematics, general topology or point-set topology is the branch of Topology which studies properties of Topological spaces and structures In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a Countable Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is In Mathematics, geometric topology is the study of Manifolds and their Embeddings Low-dimensional topology, concerning questions of dimensions A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, knot theory is the area of Topology that studies mathematical knots While inspired by knots which appear in daily life in shoelaces

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. This is a glossary of some terms used in the branch of Mathematics known as Topology. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity.

History

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

The branch of mathematics now called topology began with the investigation of certain questions in geometry. Leonhard Euler's 1736 paper on Seven Bridges of Königsberg is regarded as one of the first topological results. Year 1736 ( MDCCXXXVI) was a Leap year starting on Sunday (link will display the full calendar of the Gregorian calendar (or a Leap year The Seven Bridges of Königsberg is a famous historical problem in mathematics

The term "Topologie" was introduced in German in 1847 by Johann Benedict Listing in Vorstudien zur Topologie, Vandenhoeck und Ruprecht, Göttingen, pp. Johann Benedict Listing ( July 25, 1808 &ndash December 24 1882) was a German Mathematician. 67, 1848. However, Listing had already used the word for ten years in correspondence. "Topology", its English form, was introduced in 1883 in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". Nature is a prominent Scientific journal, first published on 4 November 1869 The term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator.

Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. Cantor, in addition to setting down the basic ideas of set theory, considered point sets in Euclidean space, as part of his study of Fourier series. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions

Henri Poincaré published Analysis Situs in 1895, introducing the concepts of homotopy and homology, which are now considered part of algebraic topology. Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician Analysis Situs is an influential Mathematical paper (and a series of addenda written by Henri Poincaré. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is

Maurice Fréchet, unifying the work on function spaces of Cantor, Volterra, Arzelà, Hadamard, Ascoli and others, introduced the metric space in 1906. Maurice Fréchet ( September 2, 1878 – June 4, 1973) was a French Mathematician. Vito Volterra ( May 3, 1860 - October 11, 1940) was an Italian Mathematician and Physicist, best known for his Cesare Arzelà (1847-1912 was an Italian Mathematician who taught at Bologna and is recognized for contributions in sequences of functions Jacques Salomon Hadamard ( December 8, 1865 – October 17, 1963) was a French Mathematician best known for his proof of In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined 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. Felix Hausdorff ( November 8, 1868 &ndash January 26, 1942) was a German Mathematician who is considered to be one of the founders In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In current usage, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski. Kazimierz Kuratowski ( Warsaw, February 2, 1896 &ndash June 18, 1980) was a Polish Mathematician and Logician

For further developments, see point-set topology and algebraic topology. In Mathematics, general topology or point-set topology is the branch of Topology which studies properties of Topological spaces and structures Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic

Elementary introduction

A continuous deformation (homotopy) of a coffee cup into a doughnut (torus) and back. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar

Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of the great unifying ideas of mathematics. General topology, or point-set topology, defines and studies properties of spaces and maps such as connectedness, compactness and continuity. In Mathematics, general topology or point-set topology is the branch of Topology which studies properties of Topological spaces and structures In Mathematics, general topology or point-set topology is the branch of Topology which studies properties of Topological spaces and structures In Mathematics, connectedness is used to refer to various properties meaning in some sense "all one piece" In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function Algebraic topology uses structures from abstract algebra, especially the group to study topological spaces and the maps between them. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

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. Kaliningrad (Калининград is a Seaport and the administrative center of Kaliningrad Oblast, the Russian Exclave between Poland 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. The Seven Bridges of Königsberg is a famous historical problem in mathematics In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects

Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair on a ball smooth. The hairy ball theorem of Algebraic topology states that there is no nonvanishing continuous Tangent vector field on the sphere " 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. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe 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 blob (subject to certain conditions on the smoothness of the surface), as long as it has no holes.

In order 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 topological equivalence. The impossibility of crossing each bridge just once applies to any arrangement of bridges topologically equivalent to those in Königsberg, and the hairy ball theorem applies to any space topologically equivalent to a sphere.

Intuitively, two spaces are topologically equivalent if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist can't tell the coffee mug out of which she is drinking from the doughnut she is eating, 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 mug is a sturdily built type of cup often used for drinking hot beverages such as Coffee, Tea, or Hot chocolate. A doughnut (also spelled "donut" is a sweet Deep-fried piece of Dough or batter.

A simple introductory exercise is to classify the lowercase letters of the English alphabet according to topological equivalence. (The lines of the letters are assumed to have non-zero width. ) In most fonts in modern use, there is a class {a, b, d, e, o, p, q} of letters with one hole, a class {c, f, h, k, l, m, n, r, s, t, u, v, w, x, y, z} of letters without a hole, and a class {i, j} of letters consisting of two pieces. g may either belong in the class with one hole, or (in some fonts) it may be the sole element of a class of letters with two holes, depending on whether or not the tail is closed. For a more complicated exercise, it may be assumed that the lines have zero width; one can get several different classifications depending on which font is used. Letter topology is of practical relevance in stencil typography: The font Braggadocio, for instance, can be cut out of a plane without falling apart. Braggadocio is a geometrically constructed sans-serif stencil Typeface designed by W

Mathematical definition

Main article: Topological space

Let X be any set and let T be a family of subsets of X. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Then T is a topology on X if

1. Both the empty set and X are elements of T.
2. Any union of arbitrarily many elements of T is an element of T.
3. Any intersection of finitely many elements of T is an element of T.

If T is a topology on X, then X together with T is called a topological space.

All sets in T are called open; note that in general not all subsets of X need be in T. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in A subset of X is said to be closed if its complement is in T (i. In Topology and related branches of Mathematics, a closed set is a set whose complement is open. e. , it is open). In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in A subset of X may be open, closed, both, or neither. In Topology, a clopen set (or closed-open set, a Portmanteau word in a Topological space is a set which is both open and closed

A function or map from one topological space to another is called continuous if the inverse image of any open set is open. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function If the function maps the real numbers to the real numbers (both space with the Standard Topology), then this definition of continuous is equivalent to the definition of continuous in calculus. In Mathematics, the real numbers may be described informally in several different ways Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives 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. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every Topological equivalence redirects here see also Topological equivalence (dynamical systems. 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 to be 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.

Some theorems in general topology

• Every closed interval in R of finite length is compact. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set More is true: In Rⁿ, a set is compact if and only if it is closed and bounded. ↔ In Topology and related branches of Mathematics, a closed set is a set whose complement is open. (See Heine-Borel theorem). In the Topology of Metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states For a Subset
• Every continuous image of a compact space is compact.
• Tychonoff's theorem: The (arbitrary) product of compact spaces is compact. In Mathematics, Tychonoff's theorem states that the product of any collection of compact Topological spaces is compact In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural
• A compact subspace of a Hausdorff space is closed.
• Every continuous bijection from a compact space to a Hausdorff space is necessarily a homeomorphism. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Topological equivalence redirects here see also Topological equivalence (dynamical systems.
• Every sequence of points in a compact metric space has a convergent subsequence. In Mathematics, a sequence is an ordered list of objects (or events
• Every interval in R is connected. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of
• Every compact m-manifold can be embedded in some Euclidean space Rⁿ. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be
• The continuous image of a connected space is connected. In Mathematics, connectedness is used to refer to various properties meaning in some sense "all one piece"
• A metric space is Hausdorff, also normal and paracompact. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In Topology and related branches of Mathematics, normal spaces, T4 spaces, T5 spaces, and T6 spaces In Mathematics, a paracompact space is a Topological space in which every Open cover admits an open locally finite refinement.
• The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric. In Topology and related areas of Mathematics, a metrizable space is a Topological space that is homeomorphic to a Metric space. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined
• 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. In Topology, the Tietze extension theorem states that if X is a Normal topological space and f: A &rarr R
• Any open subspace of a Baire space is itself a Baire space. In Mathematics, a Baire space is a Topological space which intuitively speaking is very large and has "enough" points for certain limit processes
• 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. The Baire category theorem is an important tool in General topology and Functional analysis. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks In Topology, a Subset A of a Topological space X is called nowhere dense if the interior of the closure of
• On a paracompact Hausdorff space every open cover admits a partition of unity subordinate to the cover. In Mathematics, a paracompact space is a Topological space in which every Open cover admits an open locally finite refinement. In Mathematics, a cover of a set X is a collection of sets such that X is a Subset of the union of sets in the collection In Mathematics, a partition of unity of a Topological space X is a set of continuous functions \{\rho_i\}_{i\in I} from X
• Every path-connected, locally path-connected and semi-locally simply connected space has a universal cover. In Mathematics, connectedness is used to refer to various properties meaning in some sense "all one piece" In Mathematics, connectedness is used to refer to various properties meaning in some sense "all one piece" In Mathematics, in particular Topology, a Topological space X is called semi-locally simply connected if every point x In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism

General topology also has some surprising connections to other areas of mathematics. For example:

• in number theory, Furstenberg's proof of the infinitude of primes. In Number theory, Hillel Furstenberg 's proof of the infinitude of primes is a celebrated topological proof that the Integers contain

Some useful notions from algebraic topology

See also list of algebraic topology topics. This is a list of Algebraic topology topics, by Wikipedia page

• Homology and cohomology: Betti numbers, Euler characteristic, degree of a continuous mapping. In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is In Mathematics, specifically in Algebraic topology, cohomology is a general term for a Sequence of Abelian groups defined from a Cochain In Algebraic topology, the Betti number of a Topological space is in intuitive terms a way of counting the maximum number of cuts that can be made without dividing In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant This article is about the term "degree" as used in algebraic topology
• Intuitively-attractive applications: Brouwer fixed-point theorem, Hairy ball theorem, Borsuk-Ulam theorem, Ham sandwich theorem. In Mathematics, the Brouwer fixed point theorem is an important Fixed point theorem that applies to finite-dimensional spaces and which forms the basis for several The hairy ball theorem of Algebraic topology states that there is no nonvanishing continuous Tangent vector field on the sphere The Borsuk–Ulam theorem states that any Continuous function from an ''n''-sphere into Euclidean ''n''-space maps some pair of Antipodal points In Measure theory, a branch of Mathematics, the ham sandwich theorem, also called the Stone–Tukey theorem after Arthur H
• Homotopy groups (including the fundamental group). In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology.
• Chern classes, Stiefel-Whitney classes, Pontryagin classes. In Mathematics, in particular in Algebraic topology and differential geometry, the Chern classes are a particular type of Characteristic class In Mathematics, the Stiefel–Whitney class arises as a type of Characteristic class associated to real vector bundles E\rightarrow X In Mathematics, the Pontryagin classes are certain Characteristic classes The Pontryagin class lies in Cohomology groups with index a multiple of four

Generalizations

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 which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories. In Mathematics, pointless topology (also called point-free or pointfree topology is an approach to Topology which avoids the mentioning of points In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' In Category theory, a branch of Mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space.

Topology in Works of Art and Literature

• Some M. C. Escher works illustrate topological concepts, such as Möbius strips and non-orientable spaces. Maurits Cornelis Escher (17 June 1898 – 27 March 1972 usually referred to as M This article is about the mathematical object See Mobius Band (music group for the music group A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back
• Both Philip K. Dick's A Scanner Darkly and Robert Anton Wilson's Schrodinger's Cat trilogy reference topological ideas. Philip Kindred Dick (December 16 – March 2) was an American Science fiction Novelist and Short story Writer. A Scanner Darkly is a 1977 Science fiction Novel by Philip K Dick. Robert Anton Wilson or RAW (born Robert Edward Wilson, January 18, 1932 &ndash January 11, 2007) was an American Schrödinger's cat is a Thought experiment, often described as a Paradox, devised by Austrian physicist Erwin Schrödinger in 1935

References

• James Munkres (1999). James Raymond Munkres is a Professor Emeritus of Mathematics at MIT and the author of several texts in the area of Topology, including Topology Topology, 2nd edition, Prentice Hall. Prentice Hall is a leading educational publisher It is an Imprint of Pearson Education Inc ISBN 0-13-181629-2.  
• John L. Kelley (1975). John Leroy Kelley ( December 6 1916, Kansas – November 26 1999, Oakland California) was an American mathematician at General Topology. Springer-Verlag. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic ISBN 0-387-90125-6.  
• Clifford A. Pickover (2006). Clifford A Pickover is an American author editor and columnist in the fields of Science, Mathematics, and Science fiction, and is employed at the 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.  
• Querenburg, Boto von, (2006), Mengentheoretische Topologie. Heidelberg: Springer-Lehrbuch. ISBN 3-540-67790-9 (German)

See also

External links

• Elementary Topology: A First Course Viro, Ivanov, Netsvetaev, Kharlamov
• Euler - A New Branch of Mathematics: Topology
• An invitation to Topology Planar Machines' web site
• Geometry and Topology Index, MacTutor History of Mathematics archive
• ODP category
• 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. The classical Mathematical puzzle known as water gas and electricity, the (three utilities problem, or sometimes the three cottage problem, can be stated In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism In Mathematics, differential topology is the field dealing with differentiable functions on Differentiable manifolds It is closely related to Differential In Mathematics, geometric topology is the study of Manifolds and their Embeddings Low-dimensional topology, concerning questions of dimensions Digital topology deals with properties and features of Two-dimensional (2D or three-dimensional (3D Digital images that correspond to topological properties Algebra Theory of equations Hisab Counterexamples in Topology (1970 2nd ed 1978 is a book on Mathematics by topologists Lynn Steen and J Link topology is the study of the linked structure of the World Wide Web. In Mathematics topological graph theory is a branch of Graph theory. This is a list of General topology topics, by Wikipedia page See also Topology glossary List of geometric topology topics This is a list of Geometric topology topics, by Wikipedia page In Formal ontology, a branch of Metaphysics, and in ontological computer science, mereotopology is a First-order theory, embodying mereological Network topology is the study of the arrangement or mapping of the elements ( links, nodes, etc This is a glossary of some terms used in the branch of Mathematics known as Topology. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. The shape of the Universe is an informal name for a subject of investigation within Physical cosmology which describes the Geometry of the Universe A topological quantum computer is a theoretical Quantum computer that employs two-dimensional Quasiparticles called Anyons whose World lines cross A topological quantum field theory (or topological field theory or TQFT) is a Quantum field theory which computes Topological invariants The Geometry Center was a Mathematics research and education center at the University of Minnesota. Hassler Whitney ( 23 March 1907 &ndash 10 May 1989) was an American Mathematician.
• "Topologically Speaking", a song about topology.
• "The Use of Topology in Dance", a review of Alvin Ailey's Memoria on ExploreDance. com in which the use of topologies as a way of structuring choreography is discussed.