Topology (from the
Greek τόπος, “place”, and λόγος, “study”) is
a major area of
mathematics concerned
with spatial properties that are preserved under
continuous deformations of objects, for
example, deformations that involve stretching, but no tearing or
gluing. It emerged through the development of concepts from
geometry and
set
theory, such as space, dimension, and transformation.
Ideas that are now classified as topological were expressed as
early as 1736, and toward the end of the 19th century, a distinct
discipline developed, which was referred to in Latin as the
geometria situs (“geometry of place”) or
analysis
situs (Greek-Latin for “picking apart of place”), and which
later acquired the modern name of topology. By the middle of the
20
^{th} century, topology had become an important area of
study within mathematics.
The word
topology is used both for the mathematical
discipline and for a family of
sets with certain properties that are used
to define a
topological space, a
basic object of topology. Of particular importance are
homeomorphisms, which can be defined as
continuous functions with a
continuous
inverse. For
instance, the function
y =
x^{3} is a
homeomorphism of the
real line.
Topology includes many subfields. The most basic and traditional
division within topology is
point-set topology, which
establishes the foundational aspects of topology and investigates
concepts inherent to topological spaces (basic examples include
compactness and
connectedness);
algebraic topology, which
generally tries to measure degrees of connectivity using algebraic
constructs such as
homotopy groups
and
homology; and
geometric
topology, which primarily studies
manifolds and their embeddings (placements) in
other manifolds. Some of the most active areas, such as
low dimensional topology and
graph theory, do not fit neatly in this
division.
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
Topology began with the investigation of certain questions in
geometry.
Euler's 1736
paper on Seven Bridges of Königsberg is regarded as one of the first academic
treatises in modern topology.
The term "Topologie" was introduced in German in 1847 by
Johann Benedict Listing in
Vorstudien zur Topologie, Vandenhoeck und Ruprecht,
Göttingen, pp. 67, 1848, who had used the word for ten years
in correspondence before its first appearance in print. "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 . However, none of
these uses corresponds exactly to the modern definition of
topology.
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
Topology, as a branch of mathematics, can be formally defined as
"the study of qualitative properties of certain objects (called
topological spaces) that are
invariant under certain kind of transformations (called
continuous maps), especially those
properties that are invariant under a certain kind of equivalence
(called
homeomorphism)."
The term
topology is also used to refer to a structure
imposed upon a set
X, a structure which essentially
'characterizes' the set
X as a
topological space by taking proper care of
properties such as
convergence,
connectedness and
continuity, upon
transformation.
Topological spaces show up naturally in almost every branch of
mathematics. This has made topology one of the great unifying ideas
of mathematics.
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 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 exact shape of the sphere; it applies to pear shapes and in
fact any kind of smooth blob, 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
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 can't distinguish a coffee mug from a doughnut, 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 precise definition of
homeomorphic, involving a continuous function with a continuous
inverse, is necessarily more technical.
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.
Image:alphabet_homeo.pngImage:alphabet_homotopy.png
An introductory exercise is to classify the uppercase letters of
the
English alphabet according to
homeomorphism and homotopy equivalence. The result depends
partially on the font used. The figures use a
sans-serif font named
Myriad. Notice that homotopy
equivalence is a rougher relationship than homeomorphism; a
homotopy equivalence class can contain several of the homeomorphism
classes. The simple case of homotopy equivalence described above
can be used here to show two letters are homotopy equivalent, e.g.
O fits inside P and the tail of the P can be squished to the "hole"
part.
Thus, the homeomorphism classes are: one hole two tails, two holes
no tail, no holes, one hole no tail, no holes three tails, a bar
with four tails (the "bar" on the
K is almost too short to
see), one hole one tail, and no holes four tails.
The homotopy classes are larger, because the tails can be squished
down to a point. The homotopy classes are: one hole, two holes, and
no holes.
To be sure we have classified the letters correctly, we not only
need to show that two letters in the same class are equivalent, but
that two letters in different classes are not equivalent. In the
case of homeomorphism, this can be done by suitably 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 some practical relevance in
stencil typography. The
font
Braggadocio, for
instance, has stencils that are made of one connected piece of
material.
Mathematical definition
Let
X be any set and let
T be a family of subsets of
X. Then
T is a
topology on
X iff
- 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 the pair (
X,
T) is
called a
topological space, and the notation
X_{T} is used to
denote a set
X endowed with the particular
topology
T.
The
open sets in X are
defined to be the members of T; 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., its complement 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 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 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.
Topology topics
Some theorems in general topology
General topology also has some surprising connections to other
areas of mathematics. For example:
Some useful notions from algebraic topology
See also
list of
algebraic topology topics.
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 art and literature
See also
References
- Bourbaki; Elements of
Mathematics: General Topology, Addison-Wesley (1966).
- Boto von Querenburg (2006). Mengentheoretische
Topologie. Heidelberg: Springer-Lehrbuch. ISBN
3-540-67790-9
- Richeson, David S. (2008) Euler's Gem: The Polyhedron
Formula and the Birth of Topology. Princeton University
Press.
External links