How is the Königsberg 7 bridge problem related to topology?

Question

How can we use topology to solve the famous konigsberg 7 bridge problem? By using graph theory we can say that there does not exists any such path but I want to know the application of topology on the 7 bridge problem. Could anybody explain it to me? Thanks

Answer 1

I don't think there's a natural solution of it using topology or any real reason to attempt to state it and solve it using tools of modern topology. The reason that this problem is commonly mentioned when talking about the history of topology and regarded as part of the begginings of topology is that problem is "topological" in the following sense:

We are talking about a "shape". A city with divided by some rivers connected by some bridges. We can easily draw this city. In this sense the problem seems geometrical, we are talking about properties of a "shape". But what differentiates it from other geometrical problems like finding perimeters, areas, or lengths is that if we bend and stretch the shape (the city, or its drawing) the answer to the problem does not change. There either is or isn't a walk that crosses all the bridges and no ammount of stretching and bending will change the answer to it.

So the problem suggests that shapes have fundamental properties that are invariant under smooth deformation (no cutting or gluing) these properties can be referred to as the topological properties of the shape. Studying these properties is the subject of topology. But this particular problem is more easily solved in terms of graph theory, it's just a simple and early example of a property that is invariant under smooth deformation; a property that is determined by some mysterious relationship between the points in a shape, a relationship that is very independent to the distance between them.