Topology
From Wikipedia, the free
encyclopedia
A Möbius strip, an
object with only one surface and one edge. Such shapes are an object of study
in topology.
Topology (from the Greek τόπος, “place”, and λόγος,
“study”) is a major area of mathematicsconcerned
with properties that are preserved under continuous deformations of objects,
such as 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. Toward the end of the 19th century, a distinct
discipline developed, which was referred to in Latin as thegeometria situs (“geometry of place”) or analysis situs (Greek-Latin for “picking
apart of place”). This later acquired the modern name of topology. By the
middle of the 20th 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.
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 andconnectedness); algebraic
topology, which generally tries to measure degrees of connectivity
using algebraic constructs such ashomotopy 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. Knot theorystudies mathematical
knots.
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. Leonhard Euler's
1736 paper on the Seven Bridges of Königsberg[1] 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,[2] who had used the word for
ten years in correspondence before its first appearance in print.
"Topology," its English form, was first used in 1883 in Listing's
obituary in the journalNature[3] 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.[citation needed] 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 establishing 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,[4] introducing the concepts of homotopy and homology, which are now considered part ofalgebraic
topology.
Maurice Fréchet,
unifying the work on function spaces of Cantor, Volterra, Arzelà, Hadamard, Ascoli, and others,
introduced the metric spacein 1906.[5] 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.[6] In current usage, a
topological space is a slight generalization of Hausdorff spaces, given in 1922
by Kazimierz
Kuratowski.[citation needed]
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)."
To put it more simply, topology is the study of continuity and connectivity.
The term topology is also used to refer to a
structure imposed upon a set X, a structure that
essentially 'characterizes' the set X as atopological 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.
A continuous deformation (a
type ofhomeomorphism)
of a mug into a doughnut (torus)
and back.
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.
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.
Equivalence classes of the English alphabet:
|
|
Homeomorphism
|
Homotopy equivalence
|
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 a set and let τ be a family of subsets of X. Then τ is called a topology on X if:
1.
Both the empty set and X are elements of τ
2.
Any union of elements of τ is an element of τ
3.
Any intersection of finitely many elements of τ is an element of τ
If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation Xτ may be used to denote a set X endowed with the particular
topology τ.
The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (i.e., its complement is
open). A subset of X may be open, closed, both (clopen set), or
neither. The empty set and X itself are always clopen.
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.
Topology topics
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. (SeeHeine–Borel
theorem).
§ Every continuous image of a
compact space 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.
§ 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 ofcountably
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:
Some useful notions from algebraic topology
§ Intuitively attractive
applications: Brouwer fixed-point theorem, Hairy
ball theorem, Borsuk–Ulam
theorem, Ham
sandwich theorem.
Generalizations
Occasionally, one needs to use the tools of
topology but a "set of points" is not available. In pointless
topology one considers instead thelattice of open sets as the basic
notion of the theory, while Grothendieck topologies are certain structures
defined on arbitrary categories that allow the definition
of sheaves on those categories, and
with that the definition of quite general cohomology theories
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