The History of Topology

Leonhard Euler is a name that is scattered all throughout mathematics. With contributions from Graph Theory to Complex Analysis, he created the notation for functions, discovered e(2.71828), and so much more. What we are going to look at today is his contribution to Topology. 

The Konigsberg Bridge Problem asks if it is possible to cross each of the seven bridges in the town only once. It remained unsolved for many years. Even though it was thought to be impossible, no mathematicians had found a way to successfully prove that so. 

Euler found interest in this problem and eventually solved it. With his solutions, the foundation for topology was laid. He proved that this problem was impossible because of the number of the odd number of bridges in relation to each of the landmasses. but instead of just proving/disproving this problem, he generalized it to other bridge problems. He determined that you must first break the problem into regions. If there is an odd number of bridges, add one and divided it by two to determine how many times it must be crossed. For example, the middle section has five bridges connecting it to other landmasses. Adding 1 yields 6, divided by 2 = 3 times that it must be entered. Once you do this for all of the landmasses, you then add it together. Following this logic, you get 3+2+2+2=9. If this was a possible bridge, that sum that we just calculated would be equal to one more than the number of bridges that there was. In this case, there are 7 bridges + 1 = 8. Our solution of 9 is more than 8 so it is possible to only cross each bridge only once.
Euler looked at this problem in terms of spaces and what separates each space. He did not look at distances in solving this problem differing it from normal geometry. This problem became the very first problem of topology.
The word topology was first used by Johann Benedict Listing in 1847. Although used before, this was the first publishing of such a topic. More topology ideas that looked at a shape's properties rather than specific measurements began to form in the mid-1800s. Listing also came up with a single-sided band at the same time that Mobius did. Then, Euler's characteristic, pertaining to 3-d polyhedra, was extended into the fourth dimension, also by listing.
New mathematicians, such as Bernhard Riemann and Camille Jordan, began to work with surfaces and their connectivity and compactness. These are all properties that are now used to define and classify spaces. Once a mathematician came up with a theory or proof for our 3-dimensions, future mathematicians would progress them to n-dimensions.
Starting with Leonhard Euler, a new way of thinking about geometry surfaces and shapes began to emerge. New theories about our world and solutions to existing problems could be found. A relatively young math field, it is one that mathematicians are still discovering and learning about today.