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

Topology (Greek topos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry. 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. Topology is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these properties are the topological invariants. ... For discussion of land surfaces themselves, see Terrain. ... Image File history File links Download high resolution version (1328x824, 295 KB) Please see the file description page for further information. ... Image File history File links Download high resolution version (1328x824, 295 KB) Please see the file description page for further information. ... A Möbius strip made with a piece of paper and tape. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... For other uses, see Geometry (disambiguation). ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

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. Of particular importance in the study of topology are functions or maps that are homeomorphisms. Informally, these functions can be thought of as those that stretch space without tearing it apart or sticking distinct parts together. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... This article is about functions in mathematics. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...

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). From around 1925 to 1975 it was an important growth area within mathematics. Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... For other uses, see Latins and Latin (disambiguation). ... For other uses, see Latins and Latin (disambiguation). ...

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 that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... The two bold paths shown above are homotopic relative to their endpoints. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... 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. This is a glossary of some terms used in the branch of mathematics known as topology. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

History

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

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. Image File history File links The seven bridges of Konigsberg - old map with bridges highlighted Public Domain Image, modified by me, released under GPL. File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links The seven bridges of Konigsberg - old map with bridges highlighted Public Domain Image, modified by me, released under GPL. File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Map of Königsberg in Eulers time showing the actual layout of the seven bridges, highlighting the river Pregolya and the bridges. ... Euler redirects here. ... Events January 26 - Stanislaus I of Poland abdicates his throne. ... Map of Königsberg in Eulers time showing the actual layout of the seven bridges, highlighting the river Pregolya and the bridges. ...

The term "Topologie" was introduced in German in 1847 by Johann Benedict Listing in Vorstudien zur Topologie, Vandenhoeck und Ruprecht, Göttingen, pp. 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". The term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Johann Benedict Listing born July 25, 1808, died December 24, 1882 was a German mathematician, born in Frankfurt, Germany, and died in Göttingen, Germany. ... Nature is a prominent scientific journal, first published on 4 November 1869. ... The Spectator is a British conservative political magazine, established 1828, published weekly. ...

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 setting down the basic ideas of set theory, considered point sets in Euclidean space, as part of his study of Fourier series. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] – January 6, 1918) was a German mathematician. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...

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 (April 29, 1854 – July 17, 1912) (IPA: [1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... Analysis Situs is an influential mathematical paper (and a series of addendums) written by Henri Poincare. ... The two bold paths shown above are homotopic relative to their endpoints. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...

Maurice Fréchet, unifying the work on function spaces of Cantor, Volterra, Arzelà, Hadamard, Ascoli and others, introduced the metric space in 1906. 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. In current usage, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski. Maurice Fréchet (born September 2, 1878, died 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 contributions to mathematical biology. ... Cesare Arzelà (1847-1912) was an Italian mathematician who taught at Bologna and is recognized for contributions in sequences of functions. ... This page is a candidate for speedy deletion. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... 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 and functional analysis. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... Kazimierz Kuratowski (born February 2, 1896, Warsaw, died June 18, 1980, Warsaw) was a Polish mathematician. ...

For further developments, see point-set topology and algebraic topology. In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...

Elementary introduction

A continuous deformation (homotopy) of a coffee cup into a doughnut (torus) and back.

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. Algebraic topology uses structures from abstract algebra, especially the group to study topological spaces and the maps between them. Image File history File links Mug_and_Torus_morph. ... Image File history File links Mug_and_Torus_morph. ... The two bold paths shown above are homotopic relative to their endpoints. ... A torus This article is about the surface and mathematical concept of a torus. ... In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... This article is about mathematics. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... This picture illustrates how the hours on a clock form a group under modular addition. ...

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. Euler redirects here. ... Kaliningrad (Russian: ; Lithuanian: Karaliaučius; German  , Polish: Królewiec; briefly Russified as Kyonigsberg), is a seaport and the administrative center of Kaliningrad Oblast, the Russian exclave between Poland and Lithuania on the Baltic Sea. ... Map of Königsberg in Eulers time showing the actual layout of the seven bridges, highlighting the river Pregolya and the bridges. ... A drawing of a graph. ...

Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair on a ball smooth." 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 blob (subject to certain conditions on the smoothness of the surface), as long as it has no holes. A failed attempt to comb a hairy ball flat, leaving two tufts at the top and bottom. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ... Vector field given by vectors of the form (−y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space. ... For other uses, see Sphere (disambiguation). ...

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. An ornamental mug A contemporary mug A mug or coffee mug is a sturdily built type of ceramic cup often used for hot beverages, such as coffee, tea, and hot chocolate. ... Doughnuts being glazed at a Krispy Kreme store in Sydney. ...

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. Abcdefghijklmnopqrstuvwxyz redirects here. ... Braggadocio is a pretentious manner. ...

Mathematical definition

Main article: Topological space

Let X be any set and let T be a family of subsets of X. Then T is a topology on X if Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

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. A subset of X is said to be closed if its complement is in T (i.e., it is open). A subset of X may be open, closed, both, or neither. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... 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. 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. 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 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. This article is about functions in mathematics. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... For other uses, see Calculus (disambiguation). ... One-to-one redirects here. ... A surjective function. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...

Some theorems in general topology

• 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 sequence of points in a compact metric space has a convergent subsequence.
• Every interval in R is connected.
• The continuous image of a connected space is connected.
• A metric space is Hausdorff, also normal and paracompact.
• 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.
• 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 mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... ↔ ⇔ ≡ logical symbols representing iff. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematical analysis, the Heine-Borel theorem, named after Eduard Heine and Émile Borel, states: A subset of the real numbers R is compact iff it is closed and bounded. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, Tychonoffs theorem states that the product of any collection of compact topological spaces is compact. ... For other senses of this word, see sequence (disambiguation). ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... This article is about mathematics. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ... In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ... 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 between elements of the set is defined. ... The Tietze extension theorem in topology states that, if X is a normal topological space and f : A → R is a continuous map from a closed subset A of X into the real numbers carrying the standard topology, then there exists a continuous map F : X → R with F(a... 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 a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In mathematics the term countable set is used to describe the size of a set, e. ... In topology, a subset A of a topological space X is called nowhere dense if the interior of the closure of A is empty. ... 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 subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X... In mathematics, a partition of unity of a topological space X is a set of continuous functions {ρi} from X to the unit interval [0,1] such that every point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all... This article is about mathematics. ... This article is about mathematics. ... In mathematics, in particular topology, a topological space X is called semi-locally simply connected if every point x in X has a neighborhood U such that the homomorphism from the fundamental group of U to the fundamental group of X, induced by the inclusion map of U into X... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open...

• in number theory, Furstenberg's proof of the infinitude of primes.

In mathematics, Hillel Furstenbergs proof of the infinitude of primes is a celebrated topological proof that the integers contain infinitely many prime numbers. ...

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.
• 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.

In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... 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 the space into two pieces. ... In algebraic topology, the Euler characteristic is a topological invariant, a number that describes one aspect of a topological spaces shape or structure. ... In mathematics, the Brouwer fixed point theorem states that every continuous function from the closed unit ball D n to itself has a fixed point. ... A failed attempt to comb a hairy ball flat, leaving two tufts at the top and bottom. ... The Borsuk-Ulam theorem states that any continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. ... The Ham sandwich theorem, also known as the Stone-Tukey theorem in topology in mathematics, states that given n objects in n-dimensional space, it is possible to divide each one in half with a single (n − 1)-dimensional hyperplane. ... The two bold paths shown above are homotopic relative to their endpoints. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Stiefel-Whitney classes arise in mathematics as a type of characteristic class associated to real vector bundles . ... In mathematics, the Pontryagin classes are certain characteristic classes. ...

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. Pointless topology is an approach to topology which avoids the mentioning of points. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... 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 open sets of a topological space. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and...

Topology in Works of Art and Literature

• Some M. C. Escher works illustrate topological concepts, such as Möbius strips and non-orientable spaces.
• Both Philip K. Dick's A Scanner Darkly and Robert Anton Wilson's Schrodinger's Cat trilogy reference topological ideas.

Maurits Cornelis Escher (June 17, 1898 – March 27, 1972), usually referred to as M. C. Escher, was a Dutch graphic artist. ... A Möbius strip made with a piece of paper and tape. ... The torus is an orientable surface. ... Philip Kindred Dick (December 16, 1928 – March 2, 1982) was an American writer, mostly known for his works of science fiction. ... A Scanner Darkly is a 1977 science fiction novel by Philip K. Dick. ... Robert Anton Wilson Robert Anton Wilson or RAW (January 18, 1932 – January 11, 2007) was a prolific American novelist, essayist, philosopher, psychologist, futurologist, anarchist, and conspiracy theory researcher. ... Schrödingers cat is a seemingly paradoxical thought experiment devised by Erwin Schrödinger that attempts to illustrate the incompleteness of the theory of quantum mechanics when going from subatomic to macroscopic systems. ...

References

• James Munkres (1999). Topology, 2nd edition, Prentice Hall. ISBN 0-13-181629-2. 
• John L. Kelley (1975). General Topology. Springer-Verlag. ISBN 0-387-90125-6. 
• Clifford A. Pickover (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. 
• Querenburg, Boto von, (2006), Mengentheoretische Topologie. Heidelberg: Springer-Lehrbuch. ISBN 3-540-67790-9 (German)

James Munkres is a professor of mathematics at MIT and the author of a major textbook, Topology. ... Pearson can mean Pearson PLC the media conglomerate. ... John Leroy Kelley (December 6, 1916 – November 26, 1999) was an American mathematician at University of California, Berkeley who worked in general topology and functional analysis. ... Springer Science+Business Media or Springer (IPA: ) is a worldwide publishing company based in Germany which focuses on academic journals and books in the fields of science, technology, mathematics, and medicine. ... Clifford A. Pickover is an author, editor, and columnist in the fields of science, mathematics, and science fiction. ...

See also

Topology Portal

Image File history File links Portal. ... The classical mathematical puzzle known as water, gas, and electricity, the (three) utilities problem, or sometimes the three cottage problem, can be stated as follows: Suppose there are three cottages on a plane (or sphere) and each needs to be connected to the gas, water, and electric companies. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... Digital topology deals with properties and features of two-dimensional (2D) or three-dimensional (3D) digital images that correspond to topological properties (e. ... This is a list of important publications in mathematics, organized by field. ... Counterexamples in Topology (1970) is a mathematics book by topologists Lynn A. Steen and J. Arthur Seebach, Jr. ... 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. ... This is a list of geometric topology topics, by Wikipedia page, organized roughly by dimension. ... Mereotopology is a formal theory, combining mereology and topology, of the topological relationships among wholes, parts, and the boundaries between parts. ... For other uses of topology, see topology (disambiguation). ... This is a glossary of some terms used in the branch of mathematics known as topology. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... The shape of the universe is a subject of investigation within cosmology. ... A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. ...

External links

Wikimedia Commons has media related to:

Topology

Wikibooks has more on the topic of

Topology

• 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.
• "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.