Why is topology useful

What is Topology? Undergraduate studies Graduate studies Employment opportunities Visitor directions. General Topology or Point Set Topology. General topology normally considers local properties of spaces, and is closely related to analysis. It generalizes the concept of continuity to define topological spaces, in which limits of sequences can be considered. Sometimes distances can be defined in these spaces, in which case they are called metric spaces; sometimes no concept of distance makes sense.

Combinatorial Topology. Combinatorial topology considers the global properties of spaces, built up from a network of vertices, edges, and faces. This is the oldest branch of topology, and dates back to Euler. It has been shown that topologically equivalent spaces have the same numerical invariant, which we now call the Euler characteristic.

Algebraic Topology. Algebraic topology also considers the global properties of spaces, and uses algebraic objects such as groups and rings to answer topological questions. Algebraic topology converts a topological problem into an algebraic problem that is hopefully easier to solve. For example, a group called a homology group can be associated to each space, and the torus and the Klein bottle can be distinguished from each other because they have different homology groups.

It depends on what you mean by "using" topology. In everyday life, you are dealing with some rudimentary topological notions subconsciusly, without actually performing any real math. In physical sciences, topology is currently sort of a hype: it has seen a high increase in research and publication volume - it will definitely yield new useful stuff, but there is also a lot of papers that are just there because it's interesting to look at something from topological perspective. Both use cases actually use a very small subset of what mathematicians call topology.

So for them, topology encountered in physics on the academic level isn't much better than counting poles on a globe. Rob Ghrist's Page: Has lots of great preprints on sensor networks, geometric and topological robotics, applied computational homology, and dynamical systems.

Especially, see his preprints on expository papers. Much of the material in Topology and its Applications on robotics is based on Rob Ghrist's work. The self-righting property is the result of the shape alone, which is determined to 0.

It is natural to want to glue two polygons only along their edges and the edges should be glued in pairs. Further, if two edges of two polygons are to be glued, they should be of the same length so we can overlap them. This tells you something about the resources you should have with you. The Odd Number Theorem.

I am not trained in general relativity. But I'll hazard See Keith McClary's comment below that this theorem implies that when observing a star we may see multiple images of it due to gravitational lensing. The theorem says that one will always observe an odd number of images.

So if you are an astronomer who found an even number of images in an observation, then either there is a flaw in the experimental setup or a hole in the laws of physics as we know it.

Topology plays a huge role in contemporary condensed matter physics, and many top-notch researchers have made a career of ite. Xiao-Liang Qi at Stanford. There are more applications of topology to condensed matter physics than I can begin to enumerate the above linked page names several , but I'll highlight two:.

A good reference for topology in physics is Mermin's paper and chapter 9 of Sethna's book. I'm not sure if this is a real-world application, but topological methods can be used for a formal proof that a system of real possibly nonlinear equations has solution.

For higher-dimensional domains, a generalization is described in this paper. Similarly, the much more complicated question of existence of a global solution of an ODE can be often resolved by topological methods. This is a very nice and readable introductory book on this topic. Check out the portal Applied Topology , which arose from Gunnar Carlsson's research group in Stanford. There are many applications areas mentioned with relations to statistics, data-mining, biology etc.

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Real life applications of Topology Ask Question. Asked 10 years ago. Active today. Viewed 82k times. What are the various real life applications of topology? Bhargav Bhargav 2, 4 4 gold badges 19 19 silver badges 25 25 bronze badges.

It has applications in diff and integro-diff and other equations. It gives the existence or maybe the uniqueness of the solution. Show 7 more comments. Active Oldest Votes.

GEdgar GEdgar Add a comment. Martin Sleziak Manos Manos Twisted K-Theory is used to classify D-branes in string theory. J J Rex Butler Rex Butler 1, 10 10 silver badges 17 17 bronze badges. Did k 27 27 gold badges silver badges bronze badges. Jonathan Pakianathan Jonathan Pakianathan 2 2 silver badges 2 2 bronze badges. Ronnie Brown Ronnie Brown I'm a graduate student yet I must admit that I know very little about the Hausdorff metric apart from a couple of times I saw it used it in measure-related theorems.

Do you have some suggestions on where I can start reading more about it? It'd be nice to see some connections to the topics you mentioned as well. Search Advanced search. Quote Post by fawazshitu » Thu Apr 11, am I studied topology for 2 semesters.

It's fun but the biggest problem i had is to prove theorems and lemmas on my own. I do understand partially When my professor teaches me but very difficult for me to prove his assignments and homework. Infact on a scale of 10, I am able to prove 2 ; Which is very bad. How can i ameliorate this situation? I also want to have ideas of the applications of Topology to real life problems. Re: Applications of Topology to real life problems. In addition to the things mentioned in the link, applications of Topology to Physics.

A comment from there: Topology is useful to mathematics as a whole. Mathematics is useful to humanity. By transitivity, Topology is useful to humanity.