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# An Analysis of Topological Ideas in Today's Mathematics

Topology Topological ideas are present in almost all areas of todays 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 Knigsberg 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 Eulers 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 Listings 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

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