Topology I INTRODUCTION Topology, branch of mathematics that explores certain properties of geometrical figures.

Topology I INTRODUCTION Topology, branch of mathematics that explores certain properties of geometrical figures. In 1930, the word topology was coined by mathematician Solomon Lefschetz. Usually classified under geometry, topology has frequently been called rubber-band, rubber-sheet, or rubber-space geometry; it deals with those properties of geometric figures in a space that remain unaltered when the space is bent, twisted, stretched, or deformed in any way; the only exceptions are that tearing the space is not allowed, and distinct points in the space cannot be made to coincide. While geometry is concerned with properties like absolute position, distance, and parallel lines, topology is concerned with properties like relative position and general shape. For example, a circle divides a flat plane into two regions, an inside and an outside. A point outside the circle cannot be connected to a point inside by a continuous path lying in the plane without crossing the circle. If the plane is deformed, it may no longer be flat or smooth, and the circle may become a crinkly curve; it will, however, maintain the property of dividing the surface into an inside and an outside. Straightness and linear and angular measure of the plane are some of the properties that are obviously not maintained if the plane is distorted. II EARLY TOPOLOGY An example of an early topological problem is the Königsberg bridge problem: Is it possible to cross the seven bridges over the Pregel River, connecting two islands and the mainland, without crossing over any bridge twice? See Fig. 1. The Swiss mathematician Leonard Euler showed that the question was equivalent to the following problem: Is it possible to draw the graph of Fig. 2 without lifting pencil from paper, and without tracing any edge twice? Euler proved that it was not possible. More generally, Euler proved that any connected linear graph, Fig. 3, for example, may be drawn with one continuous stroke without retracing edges if and only if the graph has either no odd vertices or just two odd vertices, where a vertice is odd if it is the endpoint of an odd number of lines. Because Fig. 2 has four odd vertices, it cannot be drawn by one continuous stroke without retracing lines. However, Fig. 3 has two odd vertices, so it is possible to draw that figure continuously without retracing edges. Later, in the 19th century, the German mathematician Johann Benedict Listing proved that a connected linear graph with 2n odd vertices can be drawn with n continuous strokes, each starting and ending at an odd vertex. III TOPOLOGY TODAY Topology is an active field in modern mathematics. One noted topological problem that was recently solved was to determine how many colors are needed to color an ordinary map so that no two regions that share the same border have the same color. In 1976 Kenneth Appel and Wolfgang Haken used a computer to prove that four colors are sufficient, no matter how large the map is, or how many regions are in the map. The theory of knots is a branch of topology that has many unsolved problems. A knot, such as those in Figs. 4a, 4b, and 4c may be thought of as a rubberized, simple closed curve that may be twisted, stretched, or otherwise deformed in ordinary three-dimensional-space, but not torn. Two knots are equivalent if one can be deformed into the other; otherwise, they are distinct. The knots pictured in Fig. 4b and Fig. 4c can be shown to be equivalent, but those in Figs. 4a and 4b are distinct. A complete set of characteristics that is sufficient to distinguish all knots is lacking, however. Two geometric figures or point sets are homeomorphic if a one-to-one correspondence that is continuous in both directions exists between them. The general problem of topology, still unsolved except for some special cases, is to discover a set of characteristics sufficient to identify homeomorphic figures, that is, a set of characteristics that will allow the determination of whether two given geometric figures or point sets are homeomorphic. Contributed By: James Singer Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.