TOPOLOGY

branch of mathematics that deals with selected properties of collections of related physical or abstract elements, specifically, those properties that endure when the collection undergoes distortion (as long as it remains intact). A lump of clay, for example, may be regarded as a collection of physical points that can be deformed (say, into a ball or a long, thin rod) without changing topologically. The influence of topology extends to almost every other branch of mathematics and to disciplines that formerly were not considered susceptible to mathematical treatment. For example, aspects of topology are closely related to symbolic logic; topology bears upon the design of mechanical devices, geographic maps, distribution networks, and systems for planning and controlling complex activities. Topology initially was classed as a type of geometry because it developed primarily from investigations of geometric topics carried out by Leonhard Euler in the 18th century and Bernhard Riemann and Henri Poincar in the 19th. A broadening influence originated in Georg Cantor's studies of number theory and set theory. After the Dutch mathematician L.E.J. Brouwer redefined the scope of topology and established its generality early in the 20th century, mathematical researches began to make use of concepts that are evidently topological, but not geometric in the older sense. Poincar's publications are commonly regarded as the first systematic treatment of the subdiscipline called algebraic topology, in which algebraic methods are applied to topological problems. A celebrated example from this field is a formula first stated by Ren Descartes in about 1635 and rediscovered in 1752 by Euler. Both men noted that for any polyhedron without holes, the number of vertices plus the number of faces minus the number of edges is 2. For a polyhedron with one hole, the same expression equals 0. These numbers are algebraic quantities that are unaffected by topological distortion; they depend only on the topological nature of the object. Many problems in graph theory (q.v.), such as the Knigsberg-bridge problem and the four-colour-map theorem, are consideredwith equal validityto be topics in algebraic topology. offshoot of geometry that originated during the 19th century and that studies those properties an object retains under deformationspecifically, bending, stretching and squeezing, but not breaking or tearing. Thus, a triangle is topologically equivalent to a circle but not to a straight line segment. Similarly, a solid cube made of modeling clay could be deformed into a ball by kneading. It could not, however, be molded into a solid torus (ring) unless a hole were bored through it or two surfaces were joined together. A solid cube is therefore not topologically equivalent to a finger ring. More precisely, if there are given two geometric objects or sets of points and if some two-way transformation (or operation or mapping) takes each point p of either set into one and only one point p of the other and if the transformation is continuous in the sense (which can be made precise) that points close to p become points close to p, then the transformation is called a homeomorphism and the two sets are said to be topologically equivalent. Topology is, then, the study of properties that remain invariant under homeomorphisms. This definition is intended to make it clear that the deformation concept has certain limitations. If two figures are given in Euclidean two-dimensional space, called 2that is, the space of ordinary plane geometryand if one of them consists of a circle tangent internally to a larger circle and the other consists of two externally tangent circles, then a homeomorphism exists that transforms one figure into the other and therefore the two figures are topologically equivalent. One figure cannot, however, be changed to the other by distortion in 2. It is possible to turn one of the circles through 180 around the common tangent line as axis, thus carrying it into three-dimensional space 3, and effect the deformation. The extra dimension may or may not be available, depending on the conditions of the problem. An internally tangent sphere in 3 could be continuously deformed to bring it to a position of external tangency by a rotation in hypothetical four-dimensional space 4, which might present no difficulty mathematically but would be impossible to achieve or even visualize in a physical application. The mathematical context may also prevent the use of an additional dimension. In any case, the deformation concept is not used or needed in defining topology. Additional reading Bradford H. Arnold, Intuitive Concepts in Elementary Topology (1962), is an intuitive approach, although some theorems are proved. R.H. Bing, Elementary Point Set Topology (1960), gives some topological examples and discusses the axiomatic approach to topology. Ryszard Engelking and Karol Sieklucki, Topology: A Geometric Approach (1992; originally published in Polish, 1986), is an undergraduate text which does not assume much mathematical background. Richard Courant and Herbert Robbins, What is Mathematics? (1941, reissued 1980), discusses aspects of several branches of mathematics; chapter 5 is devoted to interesting theorems and problems in topology. In somewhat the same intuitive style but devoted to a single topic is the work by H.B. Griffiths, Surfaces, 2nd ed. (1981). Lynn A. Steen and J. Arthur Seebach, Jr., Counterexamples in Topology, 2nd ed. (1978), contains examples prepared by teams of students working under the authors' direction. W.G. Chinn and N.E. Steenrod, First Concepts in Topology (1966), is also useful.Intermediate-level books suitable for those with the background of a course in calculus include John D. Baum, Elements of Point Set Topology (1964, reprinted 1991); Michael C. Gemignani, Elementary Topology, 2nd ed. (1972, reprinted 1990); Robert H. Kasriel, Undergraduate Topology (1971); Kazimierz Kuratowski, Introduction to Set Theory and Topology, completely rev. ed., trans. from Polish (1972; originally published in Polish, 1955); and M.H.A. Newman, Elements of the Topology of Plane Sets of Points, 2nd ed. (1951, reissued 1985). The first three of these are especially suitable as texts for an undergraduate course in topology. Newman's book is a classic elementary treatment of many of the topological properties of the Euclidean plane. Waclaw Sierpinski, General Topology, 2nd ed., translated from Polish (1956, reprinted 1961), was the text that introduced many graduate students to the study of topology. John L. Kelley, General Topology (1955, reprinted 1975), followed as a widely used graduate text, treating general topology in a way particularly useful to those working in analysis. Although Dick Wick Hall and Guilford L. Spencer II, Elementary Topology (1955), is labeled elementary, it has a careful treatment of some of the difficult properties of the plane. John G. Hocking and Gail S. Young, Topology (1961, reprinted 1988), has been widely used as a graduate text and not only treats general topology but also introduces many geometric ideas useful in the study of manifolds. James R. Munkres, Topology (1975), has become a standard introductory text at the advanced undergraduate and beginning graduate levels. Additional texts include Ronald Brown, Topology, rev., updated, and expanded ed. (1988); and C. Wayne Patty, Foundations of Topology (1993). Sources assuming some background in introductory topology include Paul R. Chernoff, A Simple Proof of Tychonoff's Theorem Via Nets, The American Mathematical Monthly, 99(10):932934 (1992); Sam B. Nadler, Jr., Continuum Theory (1992); Kiiti Morita and Jun-ita Nagata, Topics in General Topology (1989); and Jan van Mill and George M. Reed (eds.), Open Problems in Topology (1990), with updates in each subsequent issue of the journal Topology and Its Applications. R.H. Bing The Editors of the Encyclopdia Britannica