Motivation for the importance of topology

Question

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?

Answer 1

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!

Answer 2

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".

Answer 3

• 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.

Answer 4

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......

Answer 5

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".

Answer 6

One useful approach to teaching a new subject is to connect it to things the students are already familiar with. For example, you mentioned that they have had some treatment of graph theory. Perhaps you could take some mathematical facts from this thesis which connects topology with graph theory.

Answer 7

It's object-oriented programming, applied to mathematics (with the application to mathematics predating and anticipating the object-oriented programing paradigm, itself). Just as you have base types, and derived types in object oriented programming, so you have base formalisms and derived formalisms in mathematics; and just as the type inheritance relation yields a stratification into an inheritance hierarchy in object-oriented programming, you have a similar stratification of mathematical formalisms.

Classical analysis, when viewed this way, stratifies naturally into an inheritance hierarchy, with each of the major classical results and concepts in classical analysis finding their proper location at the appropriate stratum. Integrals (and sums) live in measure spaces, with measure theory, the differentiation operator lives in Banach spaces, parallelism lives in Affine geometry, as does linearity, the underlying formalism for partial differential equations (and such results as finding all the symmetries of a given system of partial differential equations) lives in Jet bundles, notions regarding continuity, connectedness, and the like, live in topological spaces and are recognized as topological concepts.

So, in the Great Sorting Out of the late 19th century and early 20th century, all the layers were recognized and formalized, and all the classical concepts and results were sorted into their respective strata. The typical way in which a layer came to be recognized was to take the classical result, whittle down the least and most general conditions that led to that result and to then take those conditions as being some or all the core of the corresponding stratum. (This process could be automated, by the way - so it may be possible to go back and redo the whole exercise by machine, starting with classical analysis, to see what results.)

Enough concepts were found to have an overlapping range of cores that centered on the notion of topology that the layer known as Topology came to be recognized in its own right. A typical starting presentation of the subject will start with a zillion different equivalent ways to define the concept of a topological space (open sets, closed sets, the interior operator, the closure operator, the boundary operator, neighborhoods, etc.); and each of those ways may have, historically, been a separate and independent way in which the concept of a topological space was first recognized.

The Intermediate Value Theorem, essentially, says that a continuous function ranges over all the points between any two values it takes, as the independent variables go from one setting to the other. That has the topological concepts of connectivity and continuity underlying it. Another theorem states that a continuous function assumes a minimum and maximum value as its independent variables range over a bounded interval. That's directly connected to the topological concept of compactness. The underlying role of each of those concepts were recognized in enough other contexts to justify calling them out, as well as calling out the "topology" stratum as a bona fide stratum in its own right.

A vestige of the historical evolution of these concepts remains intact in most presentations - the portfolio of examples (or exercises) typically displayed, each time a topological concept or result is rolled out, during the presentation ... because humans are creatures of habit and always keep harping the same notions.

Within the inheritance hierarchy, the "topology" stratum serves as a base formalism for other derived formalisms, including "measure theory", "Banach space", "manifold theory" and so on. So, the net result of The Great Sorting Out was an hierarchy of (now) well-established post-classical 19-20th century theories, which topology is at the relatively low end of.

Beneath it is a deeper, even more base, formalism: that of the point set, with its corresponding formalism of set theory - so the two are often presented together. However, it is possible to divorce the "topology" stratum from the underlying "point set" stratum and treat it autonomously - which leads to "pointless topology". In pointless topology, the basic objects are not points, at all, but the hierarchy of open sets - with the sets, themselves, taken as the fundamental objects, paying no regard to what their respective constituencies are.

Pointless Toplology: https://en.wikipedia.org/wiki/Pointless_topology

That tends to be less a form of analysis and more an algebraic theory. A lot of the proofs of results in topology can be done without making reference to open sets as being sets of points, but just objects in their own right. So, there's enough there to serve as the basis of another under-stratum to topology. In that respect, the "topology" layer has a dual inheritance from two separate base formalisms: the "pointless topology" formalism and the "set theory" formalism.