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Topology

(topology)





Topology is the study or science of places. It derives its name from the Greek words τοπος meaning place and λογος meaningstudy, talk.

See also: earth science , geography , human geography , geomorphology


In architecture , topology is a term used todescribe spatial effects which can not be described by topography , i.e.,social, economical, spatial or phenomenological interactions.


In mathematics , topology is a branch concerned with thestudy of topological spaces . (The term topology is alsoused for a set of open sets used to define topological spaces, but this article focuses on the branch of mathematics. Wiring and computer network topologies are discussed in network topology .)

Topology is also concerned with the study of the so-called topological properties of figures, that is to say properties thatdo not change under bicontinuous one-to-one transformations (called homeomorphisms ). Two figures that can be deformed one into the other are called homeomorphic, and areconsidered to be the same from the topological point of view. For example a solid cube and a solid sphere are homeomorphic.

However, it is not possible to deform a sphere into a circle by a bicontinuous one-to-one transformation. Dimension is infact, a topological property. In a sense, topological properties are the deeper properties of figures.

The topology glossary contains definitions of terms usedthroughout topology.

Contents

History

The root of topology was in the study of geometry in ancient cultures. Leonhard Euler 's 1736 paper on Seven Bridges of Königsberg is regarded as oneof the first results on geometry that does not depend on any measurements, i.e., one of the first topological results.

Georg Cantor , 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.

Maurice Fréchet , unifying the work on function spaces of Cantor,Volterra, Arzelŕ, Hadamard , Ascoli and others, introduced theconcept of metric space in 1906.

In 1914, Felix Hausdorff , generalizing the notion of metric space, coined theterm "topological space" and gave the definition for what is now called Hausdorff space .

Finally, a further slight generalization in 1922, by Kuratowski, gives the present-day concept of topological space.

Elementary introduction

Topological spaces show up naturally in mathematicalanalysis , abstract algebra and geometry . This has made topology one of the great unifying ideas of mathematics. General topology , or point-set topology, defines and studiessome useful properties of spaces and maps, such as connectedness , compactness and continuity . Algebraic topology is a powerful tool to studytopological spaces, and the maps between them. It associates "discrete", more computable invariants to maps and spaces, often ina 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 LeonhardEuler, 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 lengthsof the bridges, nor on their distance from one another, but only on connectivity properties: which bridges are connected to whichislands or riverbanks. This problem, the Seven Bridges of Königsberg , is now a famous problem in introductory mathematics.

Similarly, the hairy ball theorem of algebraic topology saysthat "one cannot comb the hair on a ball smooth". This fact is immediately convincing to most people, even though they might notrecognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere . As with the Bridges ofKönigsberg, the result does not depend on the exact shape of the sphere; it applies to pear shapes and in fact any kind ofblob (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 whatproperties these problems do rely on. From this need arises the notion of topological equivalence. Theimpossibility of crossing each bridge just once applies to any arrangement of bridges topologically equivalent to those inKönigsberg, and the hairy ball theorem applies to any space topologically equivalent to a sphere. Formally, two spaces aretopologically equivalent if there is a homeomorphism between them. Inthat case the spaces are said to be homeomorphic, and they are considered to be essentially the same for the purposes oftopology.

Formally, a homeomorphism is defined as a continuous bijection with a continuous inverse , which is not terribly intuitive even to one who knows what the words in the definition mean. Amore informal criterion gives a better visual sense: two spaces are topologically equivalent if one can be deformed into theother without cutting it apart or gluing pieces of it together. The traditional joke is that the topologist can't tell the coffeecup she is drinking out of from the donut she is eating, since a sufficiently pliable donut could be reshaped to the form of acoffee 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 according to topological equivalence. To be simple, it is assumed that the lines of theletters have nonzero width. Then 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 eitherbelong in the class with one hole, or be the sole element of a class of letters with two holes, depending on whether or not thetail is closed. For a more complicated exercise, it may be assumed that the lines have zero width; one can get several differentclassifications depending on which font is used.

Some theorems in general topology

Some useful notions from algebraic topology

See also list of algebraictopology topics .

Outline of the deeper theory

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

Related articles

See also

Importantpublications in topology

External links


Topics in mathematics related to structure

Abstract algebra | Number theory | Algebraic geometry | Group theory | Monoids | Analysis |Topology | Linear algebra | Graph theory | Universal algebra | Category theory | Order theory


Topics in mathematics related to spaces

Topology | Geometry | Trigonometry | Algebraic geometry | Differential geometry and topology | Algebraic topology | Linear algebra | Fractal geometry | Compact space



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