Essay, Research Paper: Topology

Topology
Topological ideas are present in almost all areas of today's mathematics. The subject of topology itself consists of several different branches, such as point set topology, algebraic topology and differential topology, which have relatively little in common. We shall trace the rise of topological concepts in a number of different situations. Perhaps the first work which deserves to be considered as the beginnings of topology is due to Euler. In 1736 Euler published a paper on the solution of the K nigsberg bridge problem entitled Solutio problematis ad geometriam situs pertinentis which translates into English as The solution of a problem relating to the geometry of position. The title itself indicates that Euler was aware that he was dealing with a different type of geometry where distance was not relevant.
The next step in freeing mathematics from being a subject about measurement was also due to Euler. In 1750 he wrote a letter to Christian Goldbach which gives Euler's famous formula for a polyhedron, v - e + f = 2, where v is the number of vertices of the polyhedron, e is the number of edges and f is the number of faces. It is interesting to realize that this, really rather simple, formula seems to have been missed by Archimedes and Descartes although both wrote extensively on polyhedra. The reason must be that to everyone before Euler, it had been impossible to think of geometrical properties without measurement being involved.
Johann Benedict Listing (1802-1882) was the first to use the word topology. Listing's topological ideas were due mainly to Gauss, although Gauss himself chose not to publish any work on topology. Listing wrote a paper in 1847 called Vorstudien zur Topologie although he had already used the word for ten years in correspondence. The 1847 paper is not very important, although he also introduces the idea of a complex, since it is extremely elementary. In 1861 Listing published a much more important paper in which he described the M bius band (4 years before M bius) and studied components of surfaces and connectivity.
A second way in which topology developed was through the generalization of the ideas of convergence. This process really began in 1817 when Bolzano removed the association of convergence with a sequence of numbers and associated convergence with any bounded infinite subset of the real numbers.
Cantor in 1872 introduced the concept of the first derived set, or set of limit points, of a set. He also defined closed subsets of the real line as subsets containing their first derived set. Cantor also introduced the idea of an open set another fundamental concept in point set topology.
There is a third way in which topological concepts entered mathematics, namely via functional analysis. This was a topic which arose from mathematical physics and astronomy, brought about because the methods of classical analysis were somewhat inadequate in tackling certain types of problems. Jacob Bernoulli and Johann Bernoulli invented the calculus of variations where the value of an integral is thought of as a function of the functions being integrated.
Jules Henri Poincar developed many of his topological methods while studying ordinary differential equations which arose from a study of certain astronomy problems. His study of autonomous systems involved looking at the totality of all solutions rather than at particular trajectories as had been the case earlier. The collection of methods developed by Poincar was built into a complete topological theory by Brouwer in 1912.