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. Geometry is concerned with properties like absolute position, distance, and parallel lines, but topology is only 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 Kِnigsberg 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 figure 1. The Swiss mathematician Leonhard Euler showed that the question was equivalent to the following problem: is it possible to draw the graph of figure 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, figure 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 figure 2 has four odd vertices, it cannot be drawn by one continuous stroke without retracing lines. However, figure 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.