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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 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 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. For instance, the function y = x3 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.

Contents

History

The Seven Bridges of Königsberg is a famous problem solved by Euler.

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.[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 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.

A continuous deformation (homeomorphism) of a coffee cup 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.

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 a topology on X iff

  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 XT 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

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Some theorems in general topology

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

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.

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

  • 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. 
  • Kelly, 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


Wikibooks

Up to date as of January 23, 2010

From Wikibooks, the open-content textbooks collection

Contents

Introduction

General Topology is based solely on set theory and concerns itself with structures of sets. It is at its core a generalization of the concept of distance, though this will not be immediately apparent for the novice student. Topology generalizes many distance related concepts, such as continuity, compactness and convergence.

For an overview of the subject of topology, please see the Wikipedia entry.

Before You Begin

In order to make things easier for you as a reader, as well as for the writers, you will be expected to be familiar with a few topics before beginning.

Point - Set Topology

Some Set Theory

Introduction to Topology

Properties of Topological Spaces

Algebraic Topology

Homotopy

Polytopes

  • Chapter 5.1 Simplicial complexes
  • Barycentric Coordinates
  • Geometric Complexes
  • Barycentric Subdivision
  • Simplical Mappings
  • Imbedding Theorem

Homology

  • Relative Homology
  • Exact Sequences
  • Mayer-Victoris Sequence
  • Eilenburg-Steenrod
  • Excision Theorem
  • Relative Homotopy
  • Cohomology
  • Cohomology Product
  • Cap-Product
  • Relative Cohomology
  • Induced Homeomorphism
  • Singular Homology
  • Victoris Homology
  • Čech Homology

Differential Topology

Help

Question & Answer

Have a question? Why not ask the very textbook that you are learning from?


1. What is the difference between topology, algebra and analysis?

  • Topology is a generalization of analysis and geometry. It comes in many flavors: point-set topology, manifold topology and algebraic topology, to name a few. All topology generalizes concepts from analysis dealing with space such as continuity of functions, connectedness of a space, open and closed sets, (etc.). Algebraic topology attributes algebraic structures (groups, rings etc.) to families of topological spaces to distinguish topological differences in those families. Manifold topology works with spaces that are locally the same as Euclidean space, ie surfaces. Often manifolds are equipped with extra structure, such as smooth, PL, symplectic, etc. A naive description of topology is that it identifies those qualities of a space that do not change under twisting and stretching of that space. As such, it is popularly referred to as "rubber sheet geometry." In reality topology does far more than this, in fact providing a rigorous foundation under all branches of mathematics dealing with "spaces."
  • Algebra deals with the structure of sets under various operations with particular properties. Commonly studied algebraic objects include Groups, Rings and Field. One of the major results from Algebra include Galois Theory, which eventually shows that there is no general solution to quartic equations. Also important results from Algebra are the Fundamental Theorem of Algebra (which says that, in the Field of Complex numbers, every polynomial has at most n roots), Group Classification, and much more.
  • Analysis (or specifically real analysis) on the other hand deals with the real numbers \mathbb{R} and the standard topology and algebraic structure of \mathbb{R}. Analysis provides rigorous proofs for the definitions of derivatives and integrals, as well as treatments of sequences and limits. One can, in some sense, view it as a rigorous treatment of the Calculus.


2. How are the concepts of base and open cover related? It seems that every base is an open cover, but not every open cover is a base. But, why are both concepts needed?

  • The terms base and open cover are not evidently related. Every base is an open cover which is probably the main relation. Take a second countable topological space for instance (second countable means that the space has a countable base for its topology). Such a space satisfies the property that every open cover has a countable sub-cover. To prove this we use the countability of the base. Basically, for any open cover, we choose for each element of the space, an element of the open cover containing it and hence a basis element contained in that element of open cover. Therefore, for any open cover, we can generate a open cover of basis elements that is an 'open refinement' (see Wikipedia for definition). From here we can get properties of open covers from properties of the base. If the base is countable, we can generate a countable open cover from the original cover.

The reason we have both definitions is because these two things have different properties. The most useful fact about a base is that it determines the topology. A basis must have "arbitrarily small" sets, that is, any open set contains a basis element. On the other hand, an open cover does not determine the topology at all. It can be used to build things such as partitions of unity, and often draws on the compactness property. Topology Expert (talk) 04:17, 8 June 2008 (UTC)

Further Reading

General Topology

Aleksandrov; Combinatorial Topology (1956)

Baker; Introduction to Topology (1991)

Dixmier; General Topology (1984)

Engelking; General Topology (1977)

Munkres; Topology (2000)

James; Topological and Uniform Spaces (1987)

Jänich; Topology (1984)

Kuratowski; Introduction to Set Theory and Topology (1961)

Kuratowski; Topology (1966)

Roseman; Elementary Topology (1999)

Seebach, Steen; Counterexamples in Topology (1978)

Willard; General Topology (1970)

Algebraic Topology

Marvin Greenberg and John Harper; Algebraic Topology (1981)

Allen Hatcher, Algebraic Topology (2002) [1]

Hu, Sze-tsen, Cohomology Theory (1968)

Hu, Sze-tsen, Homology Theory (1966)

Hu, Sze-tsen, Homotopy Theory (1959)

Albert T. Lundell and Stephen Weingram, The Topology of CW Complexes (1969)

Joerg Mayer, Algebraic Topology (1972)

James Munkres, Elements of Algebraic Topology (1984)

Joseph J. Rotman, An Introduction to Algebraic Topology (1988)

Edwin Spanier, Algebraic Topology (1966)

External Links


Simple English

Topology is the study of how spaces are organized, how the objects are structured in terms of position. It also studies how spaces are connected. It is divided into algebraic topology, differential topology and geometric topology.

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

Topology has sometimes been called rubber-sheet geometry, because in topology there is no difference between a circle and a square (a circle made out of a rubber band can be stretched into a square) but there is a difference between a circle and a figure eight (you cannot stretch a figure eight into a circle without tearing). The spaces studied in topology are called topological spaces. They vary from familiar manifolds to some very exotic constructions.

Natural Origin

In many problems, we often divide a large space into smaller areas, for instance, a house is divided into rooms, a nation into states, a type of quantity into numbers, etc. Each of these smaller areas (house, state, number) is next to other small areas (other houses/states/numbers), and the places where the areas meet are connections. If we write down on paper a list of spaces, and the connections between them, we have written down a description of a space -- a topological space. All topological spaces have the same properties (connections, etc.) and are made of the same structure (a list of smaller areas). This makes it easier to study how spaces behave in general, and to use general algorithms. For instance, to program a robot to navigate a house, we simply give it a list of rooms, the connections between each room (doors, etc.), and an algorithm that can determine the sequence of rooms to travel through to reach any other desired room.

We can go further by creating subdivisions of subdivisions of space. For instance, a nation divided into states, divided into counties, divided into city boundaries, etc. All this kind of information can be described using topology.


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