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Topology

Topology is a branch of mathematics Mathematics

Mathematics is the discipline that deals with concepts such as quantity [i], structure [i], space [i] a ... 

 concerned with spatial properties preserved under bicontinuous deformation ; these are the topological invariants. When the discipline was first properly founded, in the early years of the 20th century 20th century

The 20th century started on 1 January [i] 1901 [i] and ended on 31 December [i] 2000 [i], according to t ... 

, it was still called geometria situs and analysis situs . From around 1925 to 1975 it was the most important growth area within mathematics. Topology also refers to a particular mathematical object studied in this area. In this sense, a topology is a family of open set Open set

In topology [i] and related fields of mathematics [i], a set [i] U is called open if, intuitively sp ... 

s which contains the empty set and the entire space.

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Topology is a branch of mathematics Mathematics

Mathematics is the discipline that deals with concepts such as quantity [i], structure [i], space [i] a ... 

 concerned with spatial properties preserved under bicontinuous deformation ; these are the topological invariants. When the discipline was first properly founded, in the early years of the 20th century 20th century

The 20th century started on 1 January [i] 1901 [i] and ended on 31 December [i] 2000 [i], according to t... 

, it was still called geometria situs and analysis situs . From around 1925 to 1975 it was the most important growth area within mathematics.

Topology also refers to a particular mathematical object studied in this area. In this sense, a topology is a family of open set Open set

In topology [i] and related fields of mathematics [i], a set [i] U is called open if, intuitively sp ... 

s which contains the empty set and the entire space. If a family of sets is in the topology, then its union must be in the topology. If a finite family of sets is in the topology, then its finite intersection must be in the topology. A set Set

In mathematics [i], a set can be thought of as any collection [i] of distinct things considered as a who ... 

 equipped with a topology is called a topological space. The remainder of this article deals with the branch of mathematics known as topology.

Topology has sometimes been called rubber-sheet geometry, because it does not distinguish between a circle and a square but does distinguish between a circle and a figure eight . The spaces studied in topology are called topological spaces. They vary from familiar manifold Manifold

A manifold is an abstract mathematical space [i] in which every point has a neighborho ... 

s to some very exotic constructions.

Topology has introduced a new geometric language . It has had a major impact on the fields of differential geometry, algebraic geometry, dynamical systems Dynamical system

A dynamical system is a concept in mathematics [i] where a fixed rule describes the time dependence of a ... 

 and partial differential equations in the large, and several complex variables. Geometry in the sense of Michael Atiyah Michael Atiyah

Sir Michael Francis Atiyah, OM [i], FRS [i] is a British [i] ... 

 and his school now includes all of this. Internally to the subject, point-set topology or general topology is the study of topological spaces without further restrictions; other areas deal with topological spaces that look more like manifold Manifold

A manifold is an abstract mathematical space [i] in which every point has a neighborho ... 

s. These include algebraic topology , geometric topology, low-dimensional topology dealing for example with knot theory Knot theory

Knot theory is a branch of topology [i] inspired by observations, as the name suggests, of common knot [i] ... 

, and differential topology. This article is a general overview of topology. For more precise mathematical definitions, see topological spaces or one of the more specialized articles listed below. The topology glossary contains definitions of terms used throughout topology.

History



The root of topology was in the study of geometry in ancient cultures. Leibniz was the first to employ the term analysus situs, later employed in the 19th century to refer to what is now known as topology. Leonhard Euler Leonhard Euler

Leonhard Euler was a Swiss [i] mathematician [i] and physicist [i]. ... 

's 1736 paper on Seven Bridges of Königsberg Seven Bridges of Königsberg

The Seven Bridges of Knigsberg is a famous solved mathematics problem inspired by an actual place and si... 

is regarded as one of the first topological results.

Georg Cantor Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor was a German mathematician who is best known as the creator of set theory [i]... 

, the inventor of set theory, had begun to study the theory of point sets in Euclidean space, in the later part of the 19th century, as part of his study of Fourier series Fourier series

The Fourier series is a mathematical [i] tool used for analyzing an arbitrary periodic function [i] ... 

.

Henri Poincaré Henri Poincaré

Jules Henri Poincar , generally known as Henri Poincar, was one of France [i]'s greatest mathematician [i]... 

 published Analysis Situs in 1895, introducing the concepts of homotopy Homotopy

In topology [i], two continuous [i] functions [i] from one topological space [i] ... 

 and homology.

Maurice Fréchet, unifying the work on function spaces of Cantor, Volterra, Arzelŕ, Hadamard Jacques Hadamard

Jacques Solomon Hadamard was a French [i] mathematician [i] best known for his proof of the prime number theorem [i]... 

, Ascoli and others, introduced the concept of metric space in 1906.

In 1914, Felix Hausdorff Felix Hausdorff

Felix Hausdorff was a German mathematician [i] who is considered to be one of the founders of modern topology [i] ... 

, generalizing the notion of metric space, coined the term "topological space" and gave the definition for what is now called Hausdorff space Hausdorff space

In topology [i] and related branches of mathematics [i], a Hausdorff space, separated space or ... 

.

Finally, a further slight generalization in 1922, by Kazimierz Kuratowski Kazimierz Kuratowski

Kazimierz Kuratowski has been one of the best-known Polish [i] mathematician [i]. ... 

, gives the present-day concept of topological space.

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 Nature

Nature, in the broadest sense, is equivalent to the natural world, physical universe, mat... 

 to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". The separate status of the topologist, a specialist in topology, was used in 1905 in the magazine Spectator The Spectator

The Spectator is a British [i]
... 

.

Elementary introduction


Topological spaces show up naturally in mathematical analysis, abstract algebra and geometry Geometry

Geometry arose as the field of knowledge dealing with spatial relationships.... 

. This has made topology one of the great unifying ideas of mathematics. General topology, or point-set topology, defines and studies some useful properties of spaces and maps, such as connectedness, compactness and continuity. Algebraic topology is a powerful tool to study topological spaces, and the maps between them. It associates "discrete", more computable invariants to maps and spaces, often in a functorial way. Ideas from algebraic topology have had strong influence on algebra and algebraic geometry.

The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the "way they are connected together". One of the first papers in topology was the demonstration, by Leonhard Euler Leonhard Euler

Leonhard Euler was a Swiss [i] mathematician [i] and physicist [i]. ... 

, that it was impossible to find a route through the town of Königsberg 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 Seven Bridges of Königsberg

The Seven Bridges of Knigsberg is a famous solved mathematics problem inspired by an actual place and si... 

, is now a famous problem in introductory mathematics, and led to the branch of mathematics known as graph theory Graph theory

In mathematics [i] and computer science [i], graph theory is the study of graphs [i], mathema ... 

.

Similarly, the hairy ball theorem Hairy ball theorem

The hairy ball theorem of algebraic topology [i] states that, in layman's terms, "one cannot comb the ha ... 

 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 Vector field

In mathematics [i] a vector field is a construction in vector calculus [i] which associates a vector [i] ... 

 on the sphere Sphere

A sphere is a perfectly symmetrical [i] geometrical [i] object. ... 

. 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 , 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. Formally, two spaces are topologically equivalent if there is a homeomorphism between them. In that case the spaces are said to be homeomorphic, and they are considered to be essentially the same for the purposes of topology.

Formally, a homeomorphism is defined as a continuous bijection Bijection

In mathematics [i], a function [i] f from a set [i] X to a set Y is said to be b ... 

 with a continuous inverse, which is not terribly intuitive even to one who knows what the words in the definition mean. A more informal criterion gives a better visual sense: two spaces are topologically equivalent if one can be deformed into the other without cutting it apart or gluing pieces of it together. The traditional joke is that the topologist can't tell the coffee cup she is drinking out of from the doughnut Doughnut

A doughnut, or donut, is a deep-fried [i] piece of dough [i] or batter [i]. ... 

 she is eating, since a sufficiently pliable donut 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.


One simple introductory exercise is to classify the lowercase letters of the English alphabet Latin alphabet

The Latin alphabet, also called the Roman alphabet, is the most widely used alphabet [i]ic writing system [i] ... 

 according to topological equivalence. To be simple, it is assumed that the lines of the letters have nonzero width. Then in most fonts in modern use, there is a class of letters with one hole, a class of letters without a hole, and a class of letters consisting of two pieces. g may either belong in the class with one hole, or 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 Braggadocio (typeface)

Braggadocio is a typeface [i] that was designed by W.A. Woolley [i] in 1930. ... 

, for instance, can be cut out of a plane without falling apart.

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. .
  • Every continuous image of a compact space is compact.
  • Tychonoff's theorem: The product Product topology

    In topology [i] and related areas of mathematics [i], a product space is the cartesian product [i] of a ... 

     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 Connected space

    In topology [i] and related branches of mathematics [i], a connected space is a topological space [i] wh ... 

    .
  • The continuous image of a connected space is connected.
  • A metric space is Hausdorff Hausdorff space

    In topology [i] and related branches of mathematics [i], a Hausdorff space, separated space or ... 

    , also normal Normal space

    In topology [i] and related branches of mathematics [i], normal spaces, T4 spaces, and T5 space ... 

     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.
  • Topological spaces have a topological dimension.

Some useful notions from algebraic topology

See also list of algebraic topology topics.
  • Homology and cohomology: Betti numbers, Euler characteristic Euler characteristic

    In algebraic topology [i], the Euler characteristic is a topological invariant [i], a number that descri ... 

    .
  • Intuitively-attractive applications: Brouwer fixed-point theorem Brouwer fixed point theorem

    In mathematics [i], the Brouwer fixed point theorem is an important fixed point theorem [i] that applies ... 

    , Borsuk-Ulam theorem, Ham sandwich theorem.
  • Homotopy Homotopy

    In topology [i], two continuous [i] functions [i] from one topological space [i] ... 

     groups .
  • Chern classes, Stiefel-Whitney classes, Pontryagin classes.

Outline of the deeper theory

  • fibre sequences: Puppe sequence, computations
  • Homotopy groups of spheres Homotopy groups of spheres

    Homotopy groups of spheres is a branch of mathematics, specifically algebraic topology [i], that attempt... 

  • Obstruction theory
  • K-theory: KO-theory, algebraic K-theory
  • Stable homotopy theory
  • Brown's representability theorem
  • bordism Cobordism

    In mathematics [i], cobordism is a relation between manifold [i]s, based on the idea of boundary [i]. ... 

  • Signatures
  • Brown-Peterson BP and Morava K-theory
  • Surgery obstructions
  • H-spaces, infinite loop spaces, A8 rings
  • Homotopy theory of affine schemes
  • Intersection cohomology

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 Category theory

In mathematics [i], category theory deals in an abstract way with mathematical structures and relationsh ... 

 which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.

References

    • Oxford English Dictionary Oxford English Dictionary

      The Oxford English Dictionary is a dictionary [i] published by the Oxford University Press [i] , an ... 

  • Querenburg, Boto von, , Mengentheoretische Topologie. Heidelberg: Springer-Lehrbuch. ISBN 3-540-67790-9

See also

  • Covering map
  • Differential topology
  • Geometric topology
  • Digital topology
  • Important publications in topology
  • Link topology
  • List of general topology topics
  • List of geometric topology topics
  • Mereotopology
  • Network topology Network topology

    A network topology is the pattern of links [i] connecting pairs of node [i]s of a [i] ... 

  • Topology glossary
  • Topological space
  • Topology of the universe Shape of the Universe

    The shape of the Universe is a subject of investigation within physical cosmology [i].... 

  • Counterexamples in Topology

External links

  • Viro, Ivanov, Netsvetaev, Kharlamov
  • Planar Machines' web site
  • ,
  • at The Geometry Center
  • Aisling McCluskey and Brian McMaster, Topology Atlas







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