Topology Essay

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Topology

Topology is the study of those properties of geometric figures that are unchanged when the shape of the figure is twisted, stretched, shrunk, or otherwise distorted without breaking. It is sometimes referred to as "rubber sheet geometry" (West 577). Topology is a basic and essential part of any post school mathematics curriculum. Johann Benedict Listing introduced this subject, while Euler is regarded as the founder of topology. Mathematicians such as August Ferdinand Möbius, Felix Christian Klein, Camille Marie Ennemond Jordan and others have contributed to this field of mathematics. The Möbius band, Klein bottle, and Jordan curve are all examples of objects commonly studied. These and other topics prove to be intricate and
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C can be straightened out to make an I, the I can be bent to make an L, and so on. O and D are topologically equivalent only to each other (West 578).

Topology has an interesting history. As a branch of mathematics, it did not spring full-blown into the minds of some mathematicians. It gradually developed as a number of mathematicians experimented with the distortion of geometric figures. In the 18th and 19th centuries, Euler distorted a map into a network and concluded that the formula v - e + f = 2, where v equals the number of vertices, e equals the number of edges, and f equals the number of faces, holds true for all solids. Then, Antoine-Jean Lhuilier attempted to classify cases in which he discovered that Euler's formula was wrong. Also, Möbius discovered a one-sided surface. Carl Friedrich Gauss, a German mathematician, explored the distortion of knots. Listing published his Census; Bernhard Riemann studied the multiplicities of the roots of equations, while Klein and Fricke developed his ideas. Gustav Kirchhoff wrote on the flow of current in the electrical networks. Many others also made contributions (Flood 105-118).

The result of all these individual efforts was the eventual development of a branch of mathematics devoted exclusively to studying the constant properties of figures under distortion. The first attempt to systematize topology was made by Henri Poincaré, a French mathematician. In 1895, he

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