# Dictionary Definition

topology

### Noun

1 topographic study of a given place (especially
the history of place as indicated by its topography); "Greenland's
topology has been shaped by the glaciers of the ice age"

2 the study of anatomy based on regions or
divisions of the body and emphasizing the relations between various
structures (muscles and nerves and arteries etc.) in that region
[syn: regional
anatomy, topographic
anatomy]

3 the branch of pure mathematics that deals only
with the properties of a figure X that hold for every figure into
which X can be transformed with a one-to-one correspondence that is
continuous in both directions [syn: analysis
situs]

4 the configuration of a communication network
[syn: network
topology]

# User Contributed Dictionary

## English

### Noun

- A branch of mathematics studying those properties of a geometric figure or solid that are not changed by stretching, bending and similar homeomorphisms.
- A collection of subsets of a topological space closed under the operations of union and intersection.
- The anatomical structure of part of the body.
- The arrangement of nodes in a communications network.
- The properties of a particular technological embodiment that are not affected by differences in the physical layout or form of its application.

#### Synonyms

#### Derived terms

#### Related terms

#### Translations

study of geometric properties that are not
changed by stretching etc.

- Croatian: topologija
- Czech: topologie
- German: Topologie
- Japanese: (, isō sūgaku), (toporojī)

anatomical structure

- Croatian: topologija

arrangements of computer nodes

- Croatian: topologija
- Czech: topologie
- German: Topologie
- Japanese: トポロジー

# Extensive Definition

distinguish topography

Topology (Greek
topos, "place," and logos, "study") is a branch of mathematics that is an
extension of geometry.
Topology begins with a consideration of the nature of space,
investigating both its fine structure and its global structure.
Topology builds on set theory,
considering both sets of points and families of sets.

The word topology is used both for the area of
study and for a family of sets with certain properties described
below that are used to define a topological
space. Of particular importance in the study of topology are
functions
or maps that are homeomorphisms.
Informally, these functions can be thought of as those that stretch
space without tearing it apart or sticking distinct parts
together.

When the discipline was first properly founded,
toward the end of the 19th
century, it was called geometria situs (Latin geometry of
place) and analysis situs (Latin analysis of
place). From around 1925 to 1975 it was an important growth area
within mathematics.

Topology is a large branch of mathematics that
includes many subfields. The most basic division within topology is
point-set
topology, which investigates such concepts as compactness,
connectedness,
and countability;
algebraic
topology, which investigates such concepts as homotopy and homology;
and geometric
topology, which studies manifolds and
their embeddings, including knot
theory.

See also: topology
glossary for definitions of some of the terms used in topology
and topological
space for a more technical treatment of the subject.

## History

The branch of mathematics now called topology
began with the investigation of certain questions in geometry.
Leonhard
Euler's 1736 paper on
Seven Bridges of Königsberg is regarded as one of the first
topological results.

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 term
topologist in the sense of a specialist in topology was used in
1905 in the magazine Spectator.

Modern topology depends strongly on the ideas of
set
theory, developed by Georg Cantor
in the later part of the 19th century. Cantor, in addition to
setting down the basic ideas of set theory, considered point sets
in Euclidean
space, as part of his study of Fourier
series.

Henri
Poincaré published Analysis
Situs in 1895, introducing the concepts of homotopy and homology,
which are now considered part of algebraic topology.

Maurice
Fréchet, unifying the work on function spaces of Cantor,
Volterra,
Arzelà,
Hadamard,
Ascoli and others, introduced the metric space
in 1906. A metric space is now considered a special case of a
general topological space. In 1914, Felix
Hausdorff coined the term "topological space" and gave the
definition for what is now called a Hausdorff
space. In current usage, a topological space is a slight
generalization of Hausdorff spaces, given in 1922 by Kazimierz
Kuratowski.

For further developments, see point-set
topology and algebraic
topology.

## Elementary introduction

Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of the great unifying ideas of mathematics. General topology, or point-set topology, defines and studies properties of spaces and maps such as connectedness, compactness and continuity. Algebraic topology uses structures from abstract algebra, especially the group to study topological spaces and the maps between them.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.

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.

Intuitively, two spaces are topologically
equivalent if one can be deformed into the other without cutting or
gluing. A traditional joke is that a topologist can't tell the
coffee
mug out of which she is drinking from the doughnut
she is eating, since a sufficiently pliable doughnut 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.

A simple introductory exercise is to classify the
lowercase letters of the English
alphabet according to topological equivalence. (The lines of
the letters are assumed to have non-zero width.) 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 (in some
fonts) it may 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.

## Mathematical definition

Let X be any set and let T be a family of subsets of X. Then T is a topology on X if- Both the empty set and X are elements of T.
- Any union of arbitrarily many elements of T is an element of T.
- Any intersection of finitely many elements of T is an element of T.

If T is a topology on X, then X together with T
is called a topological space.

All sets in T are called open; note that
in general not all subsets of X need be in T. A subset of X is said
to be closed if its
complement is in T (i.e., it is open). A subset
of X may be open, closed, both, or
neither.

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 space 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 to be
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.

## 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. (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 continuous bijection from a compact space to a Hausdorff space is necessarily a homeomorphism.
- Every sequence of points in a compact metric space has a convergent subsequence.
- Every interval in R is connected.
- Every compact m-manifold can be embedded in some Euclidean space Rn.
- 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.
- Any open subspace of a Baire space is itself a Baire 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.

General topology also has some surprising
connections to other areas of mathematics. For example:

- in number theory, Furstenberg's proof of the infinitude of primes.

## Some useful notions from algebraic topology

See also list of algebraic topology topics.- Homology and cohomology: Betti numbers, Euler characteristic, degree of a continuous mapping.
- Intuitively-attractive applications: Brouwer fixed-point theorem, Hairy ball theorem, Borsuk-Ulam theorem, Ham sandwich theorem.
- Homotopy groups (including the fundamental group).
- Chern classes, Stiefel-Whitney classes, Pontryagin classes.

## 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.## Topology in Works of Art and Literature

- Some M. C. Escher works illustrate topological concepts, such as Möbius strips and non-orientable spaces.
- Both Philip K. Dick's A Scanner Darkly and Robert Anton Wilson's Schrodinger's Cat trilogy reference topological ideas.

## References

- Querenburg, Boto von, (2006), Mengentheoretische Topologie. Heidelberg: Springer-Lehrbuch. ISBN 3-540-67790-9

## See also

- Water, gas, and electricity, a classical puzzle
- Covering map
- Differential topology
- Geometric topology
- Digital topology
- Important publications in topology
- Link topology
- Topological graph theory
- List of general topology topics
- List of geometric topology topics
- Mereotopology
- Network topology
- Topology glossary
- Topological space
- Topology of the universe
- Topological Quantum Computing
- Topological quantum field theory

## External links

- Elementary Topology: A First Course Viro, Ivanov, Netsvetaev, Kharlamov
- Euler - A New Branch of Mathematics: Topology
- 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
- Moscow 1935: Topology moving towards America, a historical essay by Hassler Whitney.
- "Topologically Speaking", a song about topology.
- "The Use of Topology in Dance", a review of Alvin Ailey's Memoria on ExploreDance.com in which the use of topologies as a way of structuring choreography is discussed.

topology in Arabic: طوبولوجيا

topology in Bulgarian: Топология

topology in Catalan: Topologia

topology in Czech: Topologie

topology in Danish: Topologi

topology in German: Topologie (Mathematik)

topology in Modern Greek (1453-):
Τοπολογία

topology in Spanish: Topología

topology in Esperanto: Topologio

topology in Persian: توپولوژی

topology in French: Topologie

topology in Galician: Topoloxía

topology in Classical Chinese: 拓撲學

topology in Korean: 위상수학

topology in Croatian: Topologija

topology in Ido: Topologio

topology in Indonesian: Topologi

topology in Icelandic: Grannfræði

topology in Italian: Topologia

topology in Hebrew: טופולוגיה

topology in Georgian: ტოპოლოგია

topology in Latvian: Topoloģija

topology in Lithuanian: Topologija

topology in Dutch: Topologie

topology in Japanese: 位相幾何学

topology in Norwegian: Topologi

topology in Polish: Topologia

topology in Portuguese: Topologia
(matemática)

topology in Russian: Топология

topology in Simple English: Topology

topology in Slovak: Topológia

topology in Slovenian: Topologija

topology in Serbian: Топологија

topology in Finnish: Topologia
(matematiikka)

topology in Swedish: Topologi

topology in Tamil: இடவியல்

topology in Thai: ทอพอลอยี

topology in Vietnamese: Tô pô

topology in Turkish: Topoloji

topology in Ukrainian: Топологія

topology in Chinese: 拓扑学