The ancient Prussian city of Königsberg contained seven bridges. According to a tradition among the residents of Königsberg, it was impossible to walk over all seven bridges without crossing one of them at least twice. That problem came to the attention of the Swiss mathematician Leonhard Euler in the mid-eighteenth century, and he decided to develop a theory that would explain the puzzle. The solution Euler devised was eventually to form the basis of an entirely new branch of mathematics known as topology.
Topology is a field of geometry that examines changes that occur in an object when it is stretched, bent, twisted, or otherwise deformed in some way. It is different from other forms of mathematics in that it does not deal with metrics (distances and angles, for example), but with shapes. It is thus primarily qualitative rather than quantitative in nature. Two forms of topology exist. One, combinatorial topology, is also known as "rubber-sheet" geometry. Problems like that of the Königsberg bridges involve a practical form of combinatorial topology known as network theory. The second form of topology is point set topology in which geometric figures are regarded as discrete subsets of a structured space. The Dutch mathematician Luitzen Egbertus Jan Brouwer (1881-1966) has since shown how the two branches of topology can be combined into a single generalized subject. Euler's research on topological problems was largely ignored for a century. In fact, the term "topology" itself was not introduced until 1847 when it was first used by the German mathematician Johann Benedict Listing in his book, Vorstudien zur topologie. Listing's book is regarded by many historians as the first systematic treatment of topology.
Two of the most famous topological figures were developed by the German mathematicians August Ferdinand Möbius (1790-1868) and Christian Felix Klein (1849-1925). Möbius described a method for making a paper strip with only one side, while Klein extended this concept to three dimensions, showing how to produce a bottle with no inside. For some historians of mathematics, the real beginnings of topology as a field of mathematics can be traced to a somewhat later work by Henri Poincaré. Poincaré's book, Analysis situs, outlines many of the concepts that are still fundamental to topological studies today. The title of Poincaré's book was originally used by Gottfried Leibniz to refer to the characteristics of figures resulting from their spatial configurations. Over the next decade, Poincaré added a series of papers that, overall, laid the foundations for modern topology.
The beginnings of point set topology can be traced to the work first of the French mathematician Maurice-René Fréchet and later to that of the German mathematician Felix Hausdorff. Fréchet laid important groundwork by describing a method for determining the distance between any two points in abstract space, thereby defining the concept of metric space. Hausdorff was born in Breslau, Germany (now Wroclaw, Poland) on November 8, 1868. He graduated from the University of Leipzig in 1891 and later taught at Leipzig, Bonn, and Griefswald. He committed suicide with his wife and her sister in 1942 to avoid being sent to a concentration camp. Hausdorff's most productive work dealt with topology and set theory. He built on Fréchet's concepts, showing how geometric spaces can be regarded as sets of points and sets of relationships among points. His book, Grundzüge der Mengenlehre (1910) is now regarded as the classic work about point-set topology.
The study of topology has mushroomed during the last seventy years. It has gradually become apparent that the concepts of topology may form a basis for unifying many aspects of mathematical theory. The American mathematician Luther Eisenhart (1876-1965), for example, showed how topology is related to differential geometry. The Polish-born American, Samuel Eilenberg (1913-1998) developed techniques for using the methods of algebra to deal with problems in topology. The Russian mathematician Pavel Sergeevich Aleksandrov (1896-1982) applied the principles of solid geometry to topology and has contributed to the field of topology known has homology. One of the most intriguing consequences of topological studies has been the development of catastrophe theory. Catastrophe theory is the attempt to describe and eventually explain processes that occur as a series of sudden, discontinuous events. In the mid-1950s, the French mathematician René Thom (1923-) found that such events can often be described by using techniques from topology. The British mathematician Erik Christopher Zeeman (1925-) has also made important contributions to the development of catastrophe theory. This application of topology is particularly exciting and important since it promises to find application in many practical problems in physics, chemistry, biology, and other sciences.
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