Topology
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Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. When the discipline was first introduced it was called analysis situs (Latin analysis of place).
Topology is concerned with the study of the so-called topological properties of figures, that is to say properties that do not change under bicontinuous one-to-one transformations (called homeomorphisms). Two figures that can be deformed one into the other are called homeomorphic, and are considered 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 in fact a topological property. In a sense, topological properties are the deeper properties of figures.
The topology glossary contains definitions of terms used throughout topology.
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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 one of 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 along with Gottlob Frege, had begun to study the theory of point sets in Euclidean space, in the later part of the 19th century.
Henri Poincaré published Analysis Situs in 1895, introducing the concepts of homotopy and homology.
Maurice Fréchet, unifying the work on function spaces of Cantor, Volterra, Arzelà, Hadamard, Ascoli and others, introduced the concept of metric space in 1906.
In 1914, Felix Hausdorff, generalizing the notion of metric space, coined the term "topological space" and gave the definition for what is now called Hausdorff space.
Finally, a further slight generalization in 1922, by Kazimierz Kuratowski, 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 print by Solomon Lefschetz in 1930 to replace the earlier name "analysis situs". The separate status of the topologist, a specialist in topology, was probably established around 1920.
Elementary introduction
Topological spaces show up naturally in mathematical analysis, abstract algebra and geometry. 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, 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.
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.
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 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 donut 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 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 {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 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.
Some theorems in general topology
- Every closed interval in R of finite length is compact. More is true: In Rn, a set is compact iff 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.
- 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.
- Nice applications: Brouwer Fixed Point Theorem, Borsuk-Ulam Theorem, Ham sandwich theorem.
- Homotopy groups (including the fundamental group).
- Chern classes, Stiefel Whitney classes, Pontrjagin classes.
Outline of the deeper theory
- (Co)fibre sequences: Puppe sequence, computations
- Homotopy groups of spheres
- Obstruction theory
- K-theory: KO-theory, algebraic K-theory
- Stable homotopy theory
- Brown's representability theorem
- (Co)bordism
- Signatures
- Brown-Peterson BP and Morava K-theory
- Surgery obstructions
- H-spaces, infinite loop spaces, A∞ 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 which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.
Related articles
- Topological space, list of general topology topics
- Geometric topology, list of geometric topology topics
- Differential topology
- Network topology
- Link topology
- Topology of the universe
- Covering map
- Mereotopology
See also
Important publications in topology
External links
- ODP category (http://dmoz.org/Science/Math/Topology/)
- The Topological Zoo (http://www.geom.uiuc.edu/zoo/) by the Geometry Center at the University of Minnesota
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