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Topology

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   A Möbius strip, an object with only one surface and one edge; such
   shapes are an object of study in topology.
   A Möbius strip, an object with only one surface and one edge; such
   shapes are an object of study in topology.

   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. 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 into point-set topology,
   which investigates such concepts as compactness, connectedness,
   countability, and algebraic topology, which investigates such concepts
   as homotopy, homology, and knot theory.

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

History

   The Seven Bridges of Königsberg is a famous problem in topology.
   The Seven Bridges of Königsberg is a famous problem in topology.

   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

   A continuous deformation (homotopy) of a coffee cup into a doughnut
   (torus).
   A continuous deformation ( homotopy) of a coffee cup into a doughnut (
   torus).

   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
   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.

   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 {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 (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
    1. Both the empty set and X are elements of T.
    2. Any union of elements of T is an element of T.
    3. 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.

   A set in T is called open. The complement of a set in T is called
   closed. If neither a set nor its complement is in T, then the set is
   neither open nor closed.

   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, 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 square and the circle 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 R^n, 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.

Some useful notions from algebraic topology

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

Outline of the deeper theory

     * Fibre bundle: Puppe sequence
     * Homotopy groups of spheres
     * Obstruction theory
     * K-theory: KO-theory, algebraic K-theory
     * Stable homotopy theory
     * Brown's representability theorem
     * (Co) bordism
     * Surgery theory
     * Signature (topology)
     * Knot theory
     * Brown-Peterson BP and Morava K-theory
     * Dehn surgery and obstructions
     * H-spaces, infinite loop spaces, A[∞] rings
     * Homotopy theory of affine schemes
     * Intersection cohomology

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.
   Retrieved from " http://en.wikipedia.org/wiki/Topology"
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