Topology (from the Greek Greek , an independent branch of the Indo-European family of languages, is the language of the Greeks. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. In its ancient form, it is the language of classical ancient Greek literature and the New Testament of τόπος, “place”, and λόγος, “study”) is a major area of mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions concerned with spatial properties that are preserved under continuous In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous". An intuitive though imprecise idea of continuity is deformations of objects, for example, deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry Geometry "Earth-measuring" is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by and set theory Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, 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 the geometria 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 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 A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In with certain properties that are used to define a topological space Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology, a basic object of topology. Of particular importance are homeomorphisms In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function (from the Greek words ὅμοιος = similar and μορφή (morphē) = shape, form) is a continuous function between two topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of, which can be defined as continuous functions In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous". An intuitive though imprecise idea of continuity is with a continuous inverse In mathematics, if ƒ is a function from a set A to a set B, then an inverse function for ƒ is a function from B to A, with the property that a round trip from A to B to A (or from B to A to B) returns each element of the initial set to itself. Thus, if an input x into the function ƒ produces an output y, then inputting y into the inverse. For instance, the function y = x³ is a homeomorphism of the real line In mathematics, the real line is the line whose points correspond to the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space. It is the Euclidean space of dimension one, and can be thought of as a vector space , a metric space, a topological space, or simply as a linear continuum.

Topology includes many subfields. The most basic and traditional division within topology is point-set topology In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them. It is distinct from other branches of topology in that the topological spaces may be very general, and do not have to be at all similar to manifolds, which establishes the foundational aspects of topology and investigates concepts inherent to topological spaces (basic examples include compactness In mathematics, more specifically general topology and metric topology, a compact space is an abstract mathematical space in which, intuitively, whenever one takes an infinite number of "steps" in the space, eventually one must get arbitrarily close to some other point of the space. Thus a closed and bounded subset of a Euclidean space and connectedness In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be split naturally into connected pieces, each piece is usually called a component); algebraic topology Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism. In many situations this is too much to hope for and it is more prudent to aim for a more modest goal, classification up to homotopy, which generally tries to measure degrees of connectivity using algebraic constructs such as homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space and homology In mathematics , homology (in Greek ὁμός homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group. See homology theory for more background, or singular homology for a concrete version for topological spaces, or group; and geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another, which primarily studies manifolds In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold. Thus a line and a circle are one-dimensional manifolds, a plane and sphere are two-dimensional manifolds, and so forth and their embeddings (placements) in other manifolds. Some of the most active areas, such as low dimensional topology In mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology and graph theory In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of vertices. A graph may be undirected, meaning, 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 Leonhard Paul Euler[citation needed] was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. His surname is pronounced /ˈɔɪlər/ OY-lər (like "Oiler") in English and [ˈɔʏlɐ] in German; the pronunciation /ˈjuːlər/ EW-lər is incorrect 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 Nature is a prominent British scientific journal, first published on 4 November 1869. It is the world's most highly cited interdisciplinary science journal. Most scientific journals are now highly specialized, and Nature is among the few journals that still publish original research articles across a wide range of scientific fields. There are many 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 The Spectator is a weekly British magazine first published on 6 July 1828. It is currently owned by the Barclay brothers, who also own The Daily Telegraph. Its principal subject areas are politics and culture. It generally takes a right-of-centre, conservative editorial line, although regular contributors such as Frank Field and Martin Bright.^{[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 Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, developed by Georg Cantor Georg Ferdinand Ludwig Philipp Cantor was a mathematician, best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the 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 In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” is used to distinguish these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity, as part of his study of Fourier series In mathematics, a Fourier series decomposes a periodic function or periodic signal into a sum of simple oscillating functions, namely sines and cosines . The study of Fourier series is a branch of Fourier analysis. Fourier series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate.

Henri Poincaré Jules Henri Poincaré (French pronunciation: [ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe]) was a French mathematician, theoretical physicist, and a philosopher of science. Poincaré is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime published Analysis Situs in 1895, introducing the concepts of homotopy In topology, two continuous functions from one topological space to another are called homotopic (Greek ὁμός = same, similar, and τόπος (tópos) = place) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. An outstanding use of homotopy is the definition of and homology In mathematics , homology (in Greek ὁμός homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group. See homology theory for more background, or singular homology for a concrete version for topological spaces, or group, which are now considered part of algebraic topology Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism. In many situations this is too much to hope for and it is more prudent to aim for a more modest goal, classification up to homotopy.

Maurice Fréchet, unifying the work on function spaces of Cantor, Volterra Vito Volterra was an Italian mathematician and physicist, best known for his contributions to mathematical biology, Arzelà, Hadamard Jacques Salomon Hadamard was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations, Ascoli, and others, introduced the metric space In mathematics, a metric space is a set where a notion of distance between elements of the set is defined 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 topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently used and discussed. It implies the. In current usage, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski Kazimierz Kuratowski was a Polish mathematician and logician.

For further developments, see point-set topology In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them. It is distinct from other branches of topology in that the topological spaces may be very general, and do not have to be at all similar to manifolds and algebraic topology Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism. In many situations this is too much to hope for and it is more prudent to aim for a more modest goal, classification up to homotopy.

Elementary introduction

Topology, as a branch of mathematics, can be formally defined as "the study of qualitative properties of certain objects (called topological spaces Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology) that are invariant under certain kind of transformations (called continuous maps In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous. An intuitive though imprecise idea of continuity is given by the common), especially those properties that are invariant under a certain kind of equivalence (called homeomorphism In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function (from the Greek words ὅμοιος = similar and μορφή (morphē) = shape, form) is a continuous function between two topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of)."

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 Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology by taking proper care of properties such as convergence The limit of a sequence is one of the oldest concepts in mathematical analysis. It provides a rigorous definition of the idea of a sequence converging towards a point called the limit, connectedness In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be split naturally into connected pieces, each piece is usually called a component 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 Leonhard Paul Euler[citation needed] was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. His surname is pronounced /ˈɔɪlər/ OY-lər (like "Oiler") in English and [ˈɔʏlɐ] in German; the pronunciation /ˈjuːlər/ EW-lər is incorrect, that it was impossible to find a route through the town of Königsberg (now Kaliningrad Kaliningrad is a seaport and the administrative center of Kaliningrad Oblast, the Russian exclave between Poland and Lithuania on the Baltic Sea. The territory, the northern part of the former East Prussia, borders on NATO and EU members Poland and Lithuania, and is geographically separated from the rest of Russia) 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 The Seven Bridges of Königsberg is a notable historical problem in mathematics. Its negative resolution by Leonhard Euler in 1735 laid the foundations of graph theory and presaged the idea of topology, is now a famous problem in introductory mathematics, and led to the branch of mathematics known as graph theory In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of vertices. A graph may be undirected, meaning.

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.

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 called a topology on X if:

1. Both the empty set and X are elements of T.
2. Any union of arbitrarily many 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 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

• 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.

See also some counter-intuitive theorems, e.g. the Banach–Tarski one.

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.
• Operations: cup product, Massey product
• 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 art and literature

• Some M. C. Escher works illustrate topological concepts, such as Möbius strips and non-orientable spaces.

See also

• List of algebraic topology topics
• List of general topology topics
• List of geometric topology topics
• List of topology topics
• Publications in topology
• Topology glossary

References

• Basener, William (2006). Topology and Its Applications (1st ed.). Wiley. ISBN 0-471-68755-3.
• Bourbaki; Elements of Mathematics: General Topology, Addison-Wesley (1966).
• Breitenberger, E. (2006). "Johann Benedict Listing". in I.M. James. History of Topology. North Holland. ISBN 978-0444823755.
• Kelley, John L. (1975). General Topology. Springer-Verlag. ISBN 0-387-90125-6.
• Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.
• Pickover, Clifford A. (2006). The Möbius Strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology. Thunder's Mouth Press (Provides a popular introduction to topology and geometry). ISBN 1-56025-826-8.
• Boto von Querenburg (2006). Mengentheoretische Topologie. Heidelberg: Springer-Lehrbuch. ISBN 3-540-67790-9 (German)
• Richeson, David S. (2008) Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press.
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

External links

• Elementary Topology: A First Course Viro, Ivanov, Netsvetaev, Kharlamov
• Topology at the Open Directory Project
• 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.

Categories: Mathematical structures | Topology

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