Topology
For other senses of this word, see topology (disambiguation). |
A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. |
Topology (
Greek topos, place and
logos, study) is a
branch of
mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the
topological invariants. When the discipline was first properly founded, in the early years of the
20th century, it was still called
geometria situs (
Latin geometry of place) and
analysis situs (
Latin analysis of place). 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 sets 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
intersection must be in the topology. A
set 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 (a circle made out of a rubber band can be stretched into a square) but does distinguish between a circle and a figure eight (you cannot stretch a figure eight into a circle without tearing). The spaces studied in topology are called
topological spaces. They vary from familiar
manifolds to some very exotic constructions.
Topology has introduced a new geometric language (
simplicial complexes,
homotopy,
cohomology,
Poincaré duality,
fibrations,
vector bundles,
sheaves,
characteristic classes,
Morse functions,
homological algebra,
spectral sequences). It has had a major impact on the fields of
differential geometry,
algebraic geometry,
dynamical systems and
partial differential equations in the large, and
several complex variables.
Geometry in the sense of
Michael Atiyah 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
manifolds. These include
algebraic topology (which grew out of
combinatorial topology),
geometric topology,
low-dimensional topology dealing for example with
knot theory, 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.
|
The Seven Bridges of Königsberg is a famous problem in topology. |
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's
1736 paper on
Seven Bridges of Königsberg is regarded as 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, as part of his study of
Fourier series.
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 1883 in the journal
Nature 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.
|
A toroid in three dimensions; A coffee cup, and a donut, are both topologically indistinguishable from this toroid. |
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, and led to the branch of mathematics known as
graph theory.
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
doughnut 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.
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Coffee cups and doughnuts |
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. Letter topology is of practical relevance in stencil typography: The font
Braggadocio, for instance, can be cut out of a plane without falling apart.
* 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.
* Topological spaces have a
topological dimension.
See also
list of algebraic topology topics.
*
Homology and
cohomology:
Betti numbers,
Euler characteristic.
* Intuitively-attractive applications:
Brouwer fixed-point theorem,
Borsuk-Ulam theorem,
Ham sandwich theorem.
*
Homotopy groups (including the
fundamental group).
*
Chern classes,
Stiefel-Whitney classes,
Pontryagin classes.
* (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 cohomologyOccasionally, 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.
Topology has been influential in
psychoanalysis through its application by
Jacques Lacan (see e.g. his
letters to
Pierre Soury) since
1962, where it serves as a description language for psychic structures. For instance, according to Lacanian psychoanalysis, the relation between
the Real,
the Imaginary, and
the Symbolic is
homeomorphic to a
Borromean Knot. Historically these formalizations of psychoanalysis can be attributed to the influence of parisian
structural mathematics, like
Nicolas Bourbaki. However, Lacan's use of topology, like his use of
algebra, does not meet the standards of
rigour normally evinced by a mathematical discipline, and should be seen more as an
analogy (the value of which is left to the reader to decide upon) than as a branch of
applied mathematics.
Oxford English Dictionary*
Boto von Querenburg: Mengentheoretische Topologie. Heidelberg, Springer-Lehrbuch, ISBN 3540677909
*
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*
Topology glossary*
Topological space*
Topology of the universe*
Counterexamples in Topology*
Elementary Topology: A First Course Viro, Ivanov, Netsvetaev, Kharlamov (St. Petersburg University)
*
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
