======================================================================
= Topology =
======================================================================
Introduction
======================================================================
In mathematics, topology (from the Greek ���ο�, 'place', and λ�γο�,
'study') is concerned with the properties of a geometric object that
are preserved under continuous deformations, such as stretching,
twisting, crumpling and bending, but not tearing or gluing.
A topological space is a set endowed with a structure, called a
'topology', which allows defining continuous deformation of subspaces,
and, more generally, all kinds of continuity. Euclidean spaces, and,
more generally, metric spaces are examples of a topological space, as
any distance or metric defines a topology. The deformations that are
considered in topology are homeomorphisms and homotopies. A property
that is invariant under such deformations is a topological property.
Basic examples of topological properties are: the dimension, which
allows distinguishing between a line and a surface; compactness, which
allows distinguishing between a line and a circle; connectedness,
which allows distinguishing a circle from two non-intersecting
circles.
The ideas underlying topology go back to Gottfried Leibniz, who in the
17th century envisioned the 'geometria situs' and 'analysis situs'.
Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron
formula are arguably the field's first theorems. The term 'topology'
was introduced by Johann Benedict Listing in the 19th century,
although it was not until the first decades of the 20th century that
the idea of a topological space was developed.
Motivation
======================================================================
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.
In one of the first papers in topology, Leonhard Euler demonstrated
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 or on
their distance from one another, but only on connectivity properties:
which bridges connect to which islands or riverbanks. This Seven
Bridges of Königsberg problem 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 flat on a hairy ball without creating a cowlick."
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 shape of the sphere; it applies to any kind of smooth
blob, as long as it has no holes.
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
rely on. From this need arises the notion of homeomorphism. The
impossibility of crossing each bridge just once applies to any
arrangement of bridges homeomorphic to those in Königsberg, and the
hairy ball theorem applies to any space homeomorphic to a sphere.
Intuitively, two spaces are homeomorphic if one can be deformed into
the other without cutting or gluing. A traditional joke is that a
topologist cannot distinguish a coffee mug from a doughnut, since a
sufficiently pliable doughnut could be reshaped to a coffee cup by
creating a dimple and progressively enlarging it, while shrinking the
hole into a handle.
Homeomorphism can be considered the most basic topological
equivalence. Another is homotopy equivalence. This is harder to
describe without getting technical, but the essential notion is that
two objects are homotopy equivalent if they both result from
"squishing" some larger object.
Equivalence classes of the English (that is, Latin) alphabet
(sans-serif)
Homeomorphism Homotopy equivalence
alt={A,R} {B} {C,G,I,J,L,M,N,S,U,V,W,Z}, {D,O} {E,F,T,Y} {H,K}, {P,Q}
{X} alt={A,R,D,O,P,Q} {B}, {C,E,F,G,H,I,J,K,L,M,N,S,T,U,V,W,X,Y,Z}
An introductory exercise is to classify the uppercase letters of the
English alphabet according to homeomorphism and homotopy equivalence.
The result depends on the font used, and on whether the strokes making
up the letters have some thickness or are ideal curves with no
thickness. The figures here use the sans-serif Myriad font and are
assumed to consist of ideal curves without thickness. Homotopy
equivalence is a coarser relationship than homeomorphism; a homotopy
equivalence class can contain several homeomorphism classes. The
simple case of homotopy equivalence described above can be used here
to show two letters are homotopy equivalent. For example, O fits
inside P and the tail of the P can be squished to the "hole" part.
Homeomorphism classes are:
* no holes corresponding with C, G, I, J, L, M, N, S, U, V, W, and Z;
* no holes and three tails corresponding with E, F, T, and Y;
* no holes and four tails corresponding with X;
* one hole and no tail corresponding with D and O;
* one hole and one tail corresponding with P and Q;
* one hole and two tails corresponding with A and R;
* two holes and no tail corresponding with B; and
* a bar with four tails corresponding with H and K; the "bar" on the
'K' is almost too short to see.
Homotopy classes are larger, because the tails can be squished down to
a point. They are:
* one hole,
* two holes, and
* no holes.
To classify the letters correctly, we must show that two letters in
the same class are equivalent and two letters in different classes are
not equivalent. In the case of homeomorphism, this can be done by
selecting points and showing their removal disconnects the letters
differently. For example, X and Y are not homeomorphic because
removing the center point of the X leaves four pieces; whatever point
in Y corresponds to this point, its removal can leave at most three
pieces. The case of homotopy equivalence is harder and requires a more
elaborate argument showing an algebraic invariant, such as the
fundamental group, is different on the supposedly differing classes.
Letter topology has practical relevance in stencil typography. For
instance, Braggadocio font stencils are made of one connected piece of
material.
History
======================================================================
Topology, as a well-defined mathematical discipline, originates in the
early part of the twentieth century, but some isolated results can be
traced back several centuries. Among these are certain questions in
geometry investigated by Leonhard Euler. His 1736 paper on the Seven
Bridges of Königsberg is regarded as one of the first practical
applications of topology. On 14 November 1750, Euler wrote to a friend
that he had realised the importance of the 'edges' of a polyhedron.
This led to his polyhedron formula, (where , , and respectively
indicate the number of vertices, edges, and faces of the polyhedron).
Some authorities regard this analysis as the first theorem, signalling
the birth of topology.
Further contributions were made by Augustin-Louis Cauchy, Ludwig
Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti.
Listing introduced the term "Topologie" in 'Vorstudien zur Topologie',
written in his native German, in 1847, having used the word for ten
years in correspondence before its first appearance in print. The
English form "topology" was used in 1883 in Listing's obituary 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'.
Their work was corrected, consolidated and greatly extended by Henri
Poincaré. In 1895, he published his ground-breaking paper on 'Analysis
Situs', which introduced the concepts now known as homotopy and
homology, which are now considered part of algebraic topology.
Topological characteristics of closed 2-manifolds
rowspan=2 | Manifold !! rowspan=2 |Euler num!! rowspan="2"
|Orientability!! colspan="3" |Betti numbers!! rowspan="2" |Torsion
coefficient (1-dim)
b2
Sphere 2 Orientable 1 0 1 none
Torus 0 Orientable 1 2 1 none
2-holed torus �2 Orientable 1 4 1 none
-holed torus (genus ) Orientable 1 1 none
Projective plane 1 Non-orientable 1 0 0
2
Klein bottle 0 Non-orientable 1 1 0 2
Sphere with cross-caps () Non-orientable 1
0 2
2-Manifold with holes and cross-caps () Non-orientable
1
0 2
Unifying the work on function spaces of Georg Cantor, Vito Volterra,
Cesare Arzel� , Jacques Hadamard, Giulio Ascoli and others, Maurice
Fréchet introduced the metric space in 1906. A metric space is now
considered a special case of a general topological space, with any
given topological space potentially giving rise to many distinct
metric spaces. In 1914, Felix Hausdorff coined the term "topological
space" and gave the definition for what is now called a Hausdorff
space. Currently, a topological space is a slight generalization of
Hausdorff spaces, given in 1922 by Kazimierz Kuratowski.
Modern topology depends strongly on the ideas of set theory, developed
by Georg Cantor in the later part of the 19th century. In addition to
establishing the basic ideas of set theory, Cantor considered point
sets in Euclidean space as part of his study of Fourier series. For
further developments, see point-set topology and algebraic topology.
Topologies on sets
====================
The term 'topology' also refers to a specific mathematical idea
central to the area of mathematics called topology. Informally, a
topology tells how elements of a set relate spatially to each other.
The same set can have different topologies. For instance, the real
line, the complex plane, and the Cantor set can be thought of as the
same set with different topologies.
Formally, let be a set and let be a family of subsets of . Then is
called a topology on if:
# Both the empty set and are elements of .
# Any union of elements of is an element of .
# Any intersection of finitely many elements of is an element of .
If is a topology on , then the pair is called a topological space.
The notation may be used to denote a set endowed with the particular
topology .
The members of are called 'open sets' in . A subset of is said to be
closed if its complement is in (that is, its complement is open). A
subset of may be open, closed, both (a clopen set), or neither. The
empty set and itself are always both closed and open. An open subset
of which contains a point is called a neighborhood of .
Continuous functions and homeomorphisms
=========================================
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 spaces 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 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.
Manifolds
===========
While topological spaces can be extremely varied and exotic, many
areas of topology focus on the more familiar class of spaces known as
manifolds. A 'manifold' is a topological space that resembles
Euclidean space near each point. More precisely, each point of an
-dimensional manifold has a neighborhood that is homeomorphic to the
Euclidean space of dimension . Lines and circles, but not figure
eights, are one-dimensional manifolds. Two-dimensional manifolds are
also called surfaces, although not all surfaces are manifolds.
Examples include the plane, the sphere, and the torus, which can all
be realized without self-intersection in three dimensions, and the
Klein bottle and real projective plane, which cannot (that is, all
their realizations are surfaces that are not manifolds).
General topology
==================
General topology is the branch of topology dealing with the basic
set-theoretic definitions and constructions used in topology. It is
the foundation of most other branches of topology, including
differential topology, geometric topology, and algebraic topology.
Another name for general topology is point-set topology.
The basic object of study is topological spaces, which are sets
equipped with a topology, that is, a family of subsets, called 'open
sets', which is closed under finite intersections and (finite or
infinite) unions. The fundamental concepts of topology, such as
'continuity', 'compactness', and 'connectedness', can be defined in
terms of open sets. Intuitively, continuous functions take nearby
points to nearby points. Compact sets are those that can be covered by
finitely many sets of arbitrarily small size. Connected sets are sets
that cannot be divided into two pieces that are far apart. The words
'nearby', 'arbitrarily small', and 'far apart' can all be made precise
by using open sets. Several topologies can be defined on a given
space. Changing a topology consists of changing the collection of open
sets. This changes which functions are continuous and which subsets
are compact or connected.
Metric spaces are an important class of topological spaces where the
distance between any two points is defined by a function called a
'metric'. In a metric space, an open set is a union of open disks,
where an open disk of radius centered at is the set of all points
whose distance to is less than . Many common spaces are topological
spaces whose topology can be defined by a metric. This is the case of
the real line, the complex plane, real and complex vector spaces and
Euclidean spaces. Having a metric simplifies many proofs.
Algebraic topology
====================
Algebraic topology is a branch of mathematics that uses tools from
algebra to study topological spaces. The basic goal is to find
algebraic invariants that classify topological spaces up to
homeomorphism, though usually most classify up to homotopy
equivalence.
The most important of these invariants are homotopy groups, homology,
and cohomology.
Although algebraic topology primarily uses algebra to study
topological problems, using topology to solve algebraic problems is
sometimes also possible. Algebraic topology, for example, allows for a
convenient proof that any subgroup of a free group is again a free
group.
Differential topology
=======================
Differential topology is the field dealing with differentiable
functions on differentiable manifolds. It is closely related to
differential geometry and together they make up the geometric theory
of differentiable manifolds.
More specifically, differential topology considers the properties and
structures that require only a smooth structure on a manifold to be
defined. Smooth manifolds are 'softer' than manifolds with extra
geometric structures, which can act as obstructions to certain types
of equivalences and deformations that exist in differential topology.
For instance, volume and Riemannian curvature are invariants that can
distinguish different geometric structures on the same smooth
manifold�that is, one can smoothly "flatten out" certain manifolds,
but it might require distorting the space and affecting the curvature
or volume.
Geometric topology
====================
Geometric topology is a branch of topology that primarily focuses on
low-dimensional manifolds (that is, spaces of dimensions 2, 3, and 4)
and their interaction with geometry, but it also includes some
higher-dimensional topology. Some examples of topics in geometric
topology are orientability, handle decompositions, local flatness,
crumpling and the planar and higher-dimensional Schönflies theorem.
In high-dimensional topology, characteristic classes are a basic
invariant, and surgery theory is a key theory.
Low-dimensional topology is strongly geometric, as reflected in the
uniformization theorem in 2 dimensions - every surface admits a
constant curvature metric; geometrically, it has one of 3 possible
geometries: positive curvature/spherical, zero curvature/flat, and
negative curvature/hyperbolic - and the geometrization conjecture (now
theorem) in 3 dimensions - every 3-manifold can be cut into pieces,
each of which has one of eight possible geometries.
2-dimensional topology can be studied as complex geometry in one
variable (Riemann surfaces are complex curves) - by the uniformization
theorem every conformal class of metrics is equivalent to a unique
complex one, and 4-dimensional topology can be studied from the point
of view of complex geometry in two variables (complex surfaces),
though not every 4-manifold admits a complex structure.
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 structures defined on arbitrary categories
that allow the definition of sheaves on those categories, and with
that the definition of general cohomology theories.
Biology
=========
Knot theory, a branch of topology, is used in biology to study the
effects of certain enzymes on DNA. These enzymes cut, twist, and
reconnect the DNA, causing knotting with observable effects such as
slower electrophoresis. Topology is also used in evolutionary biology
to represent the relationship between phenotype and genotype.
Phenotypic forms that appear quite different can be separated by only
a few mutations depending on how genetic changes map to phenotypic
changes during development. In neuroscience, topological quantities
like the Euler characteristic and Betti number have been used to
measure the complexity of patterns of activity in neural networks.
Computer science
==================
Topological data analysis uses techniques from algebraic topology to
determine the large scale structure of a set (for instance,
determining if a cloud of points is spherical or toroidal). The main
method used by topological data analysis is to:
# Replace a set of data points with a family of simplicial complexes,
indexed by a proximity parameter.
# Analyse these topological complexes via algebraic topology -
specifically, via the theory of persistent homology.
# Encode the persistent homology of a data set in the form of a
parameterized version of a Betti number, which is called a barcode.
Several branches of programming language semantics, such as domain
theory, are formalized using topology. In this context, Steve Vickers,
building on work by Samson Abramsky and Michael B. Smyth,
characterizes topological spaces as Boolean or Heyting algebras over
open sets, which are characterized as semidecidable (equivalently,
finitely observable) properties.
Physics
=========
Topology is relevant to physics in areas such as condensed matter
physics, quantum field theory and physical cosmology.
The topological dependence of mechanical properties in solids is of
interest in disciplines of mechanical engineering and materials
science. Electrical and mechanical properties depend on the
arrangement and network structures of molecules and elementary units
in materials. The compressive strength of crumpled topologies is
studied in attempts to understand the high strength to weight of such
structures that are mostly empty space. Topology is of further
significance in Contact mechanics where the dependence of stiffness
and friction on the dimensionality of surface structures is the
subject of interest with applications in multi-body physics.
A topological quantum field theory (or topological field theory or
TQFT) is a quantum field theory that computes topological invariants.
Although TQFTs were invented by physicists, they are also of
mathematical interest, being related to, among other things, knot
theory, the theory of four-manifolds in algebraic topology, and to the
theory of moduli spaces in algebraic geometry. Donaldson, Jones,
Witten, and Kontsevich have all won Fields Medals for work related to
topological field theory.
The topological classification of Calabi-Yau manifolds has important
implications in string theory, as different manifolds can sustain
different kinds of strings.
In cosmology, topology can be used to describe the overall shape of
the universe. This area of research is commonly known as spacetime
topology.
Robotics
==========
The possible positions of a robot can be described by a manifold
called configuration space. In the area of motion planning, one finds
paths between two points in configuration space. These paths represent
a motion of the robot's joints and other parts into the desired pose.
Games and puzzles
===================
Tanglement puzzles are based on topological aspects of the puzzle's
shapes and components.
Fiber art
===========
In order to create a continuous join of pieces in a modular
construction, it is necessary to create an unbroken path in an order
which surrounds each piece and traverses each edge only once. This
process is an application of the Eulerian path.
See also
======================================================================
* Equivariant topology
* List of algebraic topology topics
* List of examples in general topology
* List of general topology topics
* List of geometric topology topics
* List of topology topics
* Publications in topology
* Topoisomer
* Topology glossary
* Topological geometry
* Topological order
Further reading
======================================================================
* Ryszard Engelking, 'General Topology', Heldermann Verlag, Sigma
Series in Pure Mathematics, December 1989, .
* Bourbaki; 'Elements of Mathematics: General Topology',
Addison-Wesley (1966).
*
*
* (Provides a well motivated, geometric account of general topology,
and shows the use of groupoids in discussing van Kampen's theorem,
covering spaces, and orbit spaces.)
* Wac�aw Sierpi�ski, 'General Topology', Dover Publications, 2000,
* (Provides a popular introduction to topology and geometry)
*
External links
======================================================================
*
* [
http://www.pdmi.ras.ru/~olegviro/topoman/index.html Elementary
Topology: A First Course] Viro, Ivanov, Netsvetaev, Kharlamov.
*
* [
http://www.geom.uiuc.edu/zoo/ The Topological Zoo] at The Geometry
Center.
* [
http://at.yorku.ca/topology/ Topology Atlas]
* [
http://at.yorku.ca/i/a/a/b/23.htm Topology Course Lecture Notes]
Aisling McCluskey and Brian McMaster, Topology Atlas.
*
[
https://web.archive.org/web/20090713073050/http://www.ornl.gov/sci/ortep/topolo
gy/defs.txt
Topology Glossary]
* [
http://www.ams.org/online_bks/hmath1/hmath1-whitney10.pdf Moscow
1935: Topology moving towards America], a historical essay by Hassler
Whitney.
License
=========
All content on Gopherpedia comes from Wikipedia, and is licensed under CC-BY-SA
License URL:
http://creativecommons.org/licenses/by-sa/3.0/
Original Article:
http://en.wikipedia.org/wiki/Topology
