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Starting from tomorrow, I will be tutoring some undergraduate students following a course in general topology. I am looking for examples motivating the importance of topology in mathematics which can be explained without too much difficulty using concepts of other areas of mathematics (or physics) they have already treated (those areas would be mainly analysis, complex analysis, linear algebra, a little graph theory, some numerical methods for maths, and classical mechanics, electromagnetism, special relativity, some QM and a little statistical physics for physics). I have tried looking around, but I have found little that would motivate me to follow such a course. Do anybody have some nice example?

Note: I will of course explain to them that without topology they'll be able to do very little advanced mathematics (e.g. functional analysis, differential geometry, ...)

EDIT: Ok, I gave as examples Tychonoff's theorem, Brower's fixed point theorem and the Jordan curve theorem.

I would like to keep this question alive, for personal interest. What are interesting (not too hard) applications of topology in other areas of mathematics?

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How about "You will not get a degree otherwise." :-) –  Asaf Karagila Feb 20 '13 at 15:55
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@AsafKaragila: Yeah, also that. But still, I'd like to give them a reason to be interested in topology other than the fact that they have to follow the course... –  Daniel Robert-Nicoud Feb 20 '13 at 16:03
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Have you seen this MathOverflow thread? –  Zev Chonoles Feb 20 '13 at 16:06
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Also, I have to admit my mind goes to the hilarious place where the word "tutoring" on the first line is replaced "torturing". –  Asaf Karagila Feb 20 '13 at 16:09
    
In an older edition of Boyce/DiPrima's Elementary Differential Equations and Boundary Value Problems (1969, 2nd edition) there was a very nice section that gave an overview of how the existence theorem can be proved in a way that is analogous to how the intermediate value theorem can be used to locate roots of algebraic and transcendental equations. A lot of books on metric spaces include this as well. Also, try googling "Nash equilibrium" and "topology". –  Dave L. Renfro Feb 20 '13 at 16:11
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5 Answers 5

Well, an idea would be to talk about results of Real Analysis and how topology generalises them, or how topological methods make their proofs much easier. You could consider the Intermediate Value Theorem for instance, or how the Lebesgue Number Lemma makes it very easy to prove that if $f: X \to Y$ is a continuous function between metric spaces $X,Y$ where $X$ is compact, then in fact $f$ is uniformly continuous with respect to the topology induced by the metric. Tychonoff's theorem might also be a nice one, or the converse of the Closed Graph Theorem. My guess is that you should connect it to metric spaces mostly, for they will be mostly familiar with that. If, of course, they are more advanced, you can consider talking about how topology shows that it is impossible to find a homeomorphism $f: [0,1] \to S^1$ or other similar constructions. I can keep going but I am afraid that it would be pointless if any of my previous suggestions is not readily utilisable. Best of luck!

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My first exposure to topology, before I even realized that it was topology, was in Zorich's Mathematical Analysis I. It was through the definition of a limit over a base.

After defining limits at a real number and at infinity in the standard way, which I was already familiar with, the author suddenly introduced a concept I'd never heard of: a base. A base, he said, was a collection $B$ of subsets of $\mathbb R$ such that:

  1. $a,b\in B\implies (\exists c\in B)\ c\subseteq A\cap B$
  2. $a\in B\wedge b\subseteq a\implies b\in B$
  3. $\emptyset\not\in B$

I don't have the book to hand, but I'm pretty sure that was all the properties. He then said that the symbol $x\to a$ represented the base "the collection of open intervals containing $a$", and $x\to +\infty$ represented the base "the collection of intervals unbounded on the right". At first I was confused, but as he defined the notion of a limit over an arbitrary base, I realized this was the solution to something that had always bugged me about limits. Limits at $a$ and limits at $\infty$ are very similar, conceptually, and yet the definitions are necessarily different because we can't measure "distance" from infinity. And here it was: the single, unified definition I'd always wanted. I became even more impressed when I realized the same notion even covered limits of sequences.

He explained that the definition came by simply observing that properties (1) through (3) were really the only ones used in proving the key properties of limits. Flipping back, I realized he was correct, and I understood: mathematical definitions are (sometimes) obtained by distilling a theory. You systematically gather up everything you used to prove the theorems, and obtain a precise description of all of the objects that satisfy those theorems (or at least, some sufficient conditions).

Most importantly, my understanding came from seeing a real application of topology in mathematics, not just some vague "intuition" of what the definition "means".

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Thank you for your answer. I have the book at hand, and the third condition is not necessary. Anyway this is indeed a good motivation. Next time I will TA for a course of topology I will probably use it. –  Daniel Robert-Nicoud Feb 21 at 21:20
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My professor's example: He used right index and thumb to form a ring, and left index and thumb to form another, interlocking the right hand ring. He asked if it was possible to deform him such that we unlock the two rings. The answer is no when he was wearing a watch. Then he took off the watch and said "now it is possible".

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Are you sure it's true? The number of times two paths "cross" is an invariant (at least under $C^\mathrm{someting}$ deformations). How doesn't that contradict your statement? –  Daniel Robert-Nicoud Mar 28 '13 at 16:09
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the point is that topology makes it easier to answer questions which are only a little less informative than many of ones natural questions, in many situations. E.g. instead of actually finding solutions to an equation, which is usually hard, one asks instead to show the existence of solutions, or to count the number of solutions in a weighted sense. Thus the intermediate value theorem provides an existence theorem in cases where the theorem of Abel tells us there is no formula in radicals for a root of a polynomial. Adding the mean value theorem we get estimates on the number of solutions. Similar two dimensional versions allow the fundamental theorem of algebra to be proved, again guaranteeing solutions to polynomial equations and counting their number. The general theory of degree of mapping generalizes this technique and even extends to infinite dimensional spaces. A nice source for that theory is Perspectives in Nonlinearity by Melvyn and Marion Berger. The theory of singularities of vector fields is another example of applying topology to questions of existence of solutions. I.e. the Euler characteristic of the sphere being 2 prohibits the existence of a never zero tangent vector field on the sphere. Cauchy's theorem also allows the concept of simple connectivity to be invoked to conclude existence of complex logarithms in certain regions. The list is long......

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  • In functional analysis, the notion of weak topology is a great example of application of "weird" (not metrizable) topology : arguments of weak compacity provide very quickly the existence in a lot of variational problems (including for example the Lagrangian formulation in classical mechanics or limited relativity).

  • The "Hairy ball" theorem has a funny consequence : at every given time there are two antipodal points at the surface of the Earth that have exactly the same temperature and pression.

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