-2

It is easy to see motivation for groups and fields, as abstractions of operations defined on integers, rationals, reals etc. and how the results from those abstractions apply to integers, reals etc.

What is the motivation for topology? Yes $\mathbb R$ is a topology, but what would be example of results that are trivial or possible by using topology that would be hard or maybe impossible using other methods such as group theory or fields or whatever that is not topology? are there some example of results that are easy by methods not in topology and hard or impossible with topology?

Here i am assuming topology as a tool to be used on set of problems, what i want to know is what type of problems are better suited in a topological framework than to let's say abstract algebra or vice versa.

I read Motivation behind topology but couldn't see what is significance of study of open sets, or an example of application from open sets to real numbers.

Community
  • 1
jimjim
  • 9,675
  • 1
    I think metric spaces are easier to motivate, and then topological spaces generalize metric spaces. One big concept in math is invariants: different subjects consider different objects which share certain invariants to be isomorphic. In topology this means that we study spaces "up to homeomorphism" or sometimes up to homotopy. If you need finer details than that, then topology alone will not solve your problem. – Ian Jul 24 '16 at 02:05
  • The definition of a continuous function is much easier to express with open sets than without them. – Andrew Dudzik Jul 24 '16 at 02:06
  • For me, the motivation for topology is that it is the basic tool for studying the shape of things, and the continuous deformations of these shapes. Isn't that enough motivation? – Alex Provost Jul 24 '16 at 02:12
  • 1
    $\mathbb{R}$ is not a topology. It can be made into a topological space by choosing a suitable collection of subsets of it. To answer your second question, any theorem about continuous functions you learned in calculus would be impossible to prove without some level of topology. – Aweygan Jul 24 '16 at 02:26
  • 1
    And as for your third question: I bet you would have a really hard time trying to use topology to prove Lagrange's theorem – Aweygan Jul 24 '16 at 02:28
  • 1
    @Aweygan , all was proved with epsilon delta, I dont recall anything being reproved with topology. – jimjim Jul 24 '16 at 02:29
  • 2
    @Arjang Topology means very different things to different people. I think you really should make your question more specific. What book are you trying to learn from? Are you trying to learn about general topology or algebraic or differential topology? What is your background? Have you taken real analysis? Have you learned about metric spaces? – Noah Olander Jul 24 '16 at 02:33
  • @Arjang The $\varepsilon$-$\delta$ definition of continuity is subsumed by the topological definition. – Aweygan Jul 24 '16 at 02:42
  • @NoahOlander : yes to real analysis, only intrdoduction to metric spaces, at least metric spaces made sense as they could be appliued to euclidean space , 25 years ago did all that, failed topology, never understood the motivation or problems it tried to solve – jimjim Jul 24 '16 at 03:41

1 Answers1

1

You study topology generally if you are interested in understanding stuff like connectivity, compactness and continuity. Topology actually aids analysis quite a bit - for instance in differential geometry we have the Atiyah-Singer index theorem.

In fact, you cannot really properly understand why or how calculus varies depending on the space without studying topology, which quantifies these notions rigorously. There are a number of more intuitive approaches to the axioms (for example neighbourhood spaces) but everything can be defined best by using the open set axioms. As for specific applications, I would advise you to read the posts here: Applications to algebraic problems.

I hope this helps!

Community
  • 1
Andrew Whelan
  • 2,138