Topology (from the Greek τόπος, "place", and λόγος, "study") is the mathematical study of shapes and spaces. It is a major area of mathematics concerned with the most basic properties of space, such as connectedness, continuity and boundary. It is the study of properties that are preserved under continuous deformations including stretching and bending, but not tearing or gluing. The exact mathematical definition is given below. Topology developed as a field of study out of geometry and set theory, through analysis of such concepts 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, 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 name topology. By the middle of the 20th century, topology had become an important area of study within mathematics.
Topology has many subfields.
- General topology establishes the foundational aspects of topology and investigates properties of topological spaces and investigates concepts inherent to topological spaces. It includes point-set topology, which is the foundational topology used in all other branches (including topics like compactness and connectedness).
- Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology and homotopy groups.
- Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.
- Geometric topology primarily studies manifolds and their embeddings (placements) in other manifolds. A particularly active area is low dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots.
Topology began with the investigation of certain questions in geometry. Leonhard Euler's 1736 paper on the 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, who had used the word for ten years in correspondence before its first appearance in print. The English form topology was first used in 1883 in Listing's obituary 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. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series.
Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet 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. Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski.
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 a certain kind of transformation (called a continuous map), 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.
Topology is also used to refer to a structure imposed upon a set X, a structure that 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. The Seven Bridges of Königsberg is 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 shape of the sphere; it applies to 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 cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to 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:|
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 the sans-serif Myriad font. Homotopy equivalence is a rougher relationship than homeomorphism; a homotopy equivalence class can contain several homeomorphism classes. The simple case of homotopy equivalence described above can be used here to show two letters are homotopy equivalent. For example, O fits inside P and the tail of the P can be squished to the "hole" part.
Homeomorphism classes are:
- no holes,
- no holes three tails,
- no holes four tails,
- one hole no tail,
- one hole one tail,
- one hole two tails,
- two holes no tail, and
- a bar with four tails (the "bar" on the K is almost too short to see).
Homotopy classes are larger, because the tails can be squished down to a point. They are:
- one hole,
- two holes, and
- no holes.
To be sure that the letters are classified correctly, we need to show that two letters in the same class are equivalent and two letters in different classes are not equivalent. In the case of homeomorphism, this can be done by 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.
One aspect of the notion of topological space is to give a convenient and general setting for the notion of continuity of functions. The most intuitive definition and most closely related to notions from analysis is in terms of neighbourhoods, but the most elegant and for many examples most useful definition of a topological space is in terms of open sets. Other equivalent definitions can be given, for example, in terms of closed sets and closure.
Let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:
- Both the empty set and X are elements of τ
- Any union of elements of τ is an element of τ
- 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 both closed and open.
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.
Some theorems in point-set 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 subset 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 finite-dimensional manifold can be embedded in some Euclidean space Rn.
- The continuous image of a connected space is connected.
- Every metric space is paracompact and Hausdorff, and thus normal.
- 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, Fürstenberg's proof of the infinitude of primes.
See also some counter-intuitive theorems, e.g. the Banach–Tarski paradox.
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, Jordan curve theorem.
- Homotopy groups (including the fundamental group).
- Chern classes, Stiefel–Whitney classes, Pontryagin classes.
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 structures defined on arbitrary categories that allow the definition of sheaves on those categories, and with that the definition of general cohomology theories.
- General topology
- List of algebraic topology topics
- List of examples in general topology
- List of general topology topics
- List of geometric topology topics
- List of topology topics
- Publications in topology
- Topology glossary
- Equivariant topology
- Euler, Leonhard, Solutio problematis ad geometriam situs pertinentis
- Listing, Johann Benedict, "Vorstudien zur Topologie", Vandenhoeck und Ruprecht, Göttingen, p. 67, 1848
- Tait, Peter Guthrie, "Johann Benedict Listing (obituary)", Nature *27*, 1 February 1883, pp. 316–317
- Poincaré, Henri, "Analysis situs", Journal de l'École Polytechnique ser 2, 1 (1895) pp. 1–123
- Fréchet, Maurice, "Sur quelques points du calcul fonctionnel", PhD dissertation, 1906
- Hausdorff, Felix, "Grundzüge der Mengenlehre", Leipzig: Veit. In (Hausdorff Werke, II (2002), 91–576)
- Rysxard Engelking, General Topology, Heldermann Verlag, Sigma Series in Pure Mathematics, December 1989, ISBN 3885380064.
- Bourbaki; Elements of Mathematics: General Topology, Addison–Wesley (1966).
- Breitenberger, E. (2006). "Johann Benedict Listing". In James, I. M. History of Topology. North Holland. ISBN 978-0-444-82375-5.
- Kelley, John L. (1975). General Topology. Springer-Verlag. ISBN 0-387-90125-6.
- Brown, Ronald (2006). Topology and Groupoids. Booksurge. ISBN 1-4196-2722-8. (Provides a well motivated, geometric account of general topology, and shows the use of groupoids in discussing van Kampen's theorem, covering spaces, and orbit spaces.)
- Waclaw Sierpinski, General Topology, Dover Publications, 2000, ISBN 0486411486
- 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. ISBN 1-56025-826-8. (Provides a popular introduction to topology and geometry)
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- The Topological Zoo at The Geometry Center
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- Moscow 1935: Topology moving towards America, a historical essay by Hassler Whitney.