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I'm preparing (to teach) my first class of undergraduate topology and I'm looking for some elementary, motivating applications of topology for the first day. We'll be following Munkres, starting with point-set topology and then ending the semester with advanced topics such as knot theory or algebraic topology. The students will not have had algebra and this might be the first course they take after introduction to proof.

I'm looking for everyday facts that are easily stated and proved nicely with topology. For instance, I can illustrate Brouwer's fixed point theorem with a map of campus: If I drop the map on the ground, there is some point on the map over the point it represents. If I rip the map into pieces, I can put the map parts all over campus so that no map point is over the point it represents.

Most "applications" that I can think of / are in Real Life Applications of Topology are too advanced for a first, introductory day.

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Some of my favorite results in topology are:

  • The Borsuk-Ulam Theorem: Given a continuous function $f:S^n\to\mathbb R^n$, there exists a point $x$ such that $f(x)=f(-x)$. My favorite result is that somewhere on the planet, there are antipodal points which have the same temperature and pressure. For this reason, this is sometimes called the Meteorologist's Theorem when $n=2$.
  • The Hairy Ball Theorem: There exists a nonvanishing vector field on $S^n$ if and only if $n$ is odd. This means that somewhere on earth, the wind is not blowing.
  • The Ham Sandwich Theorem: Given $n$ measurable subsets $\{A_i\}$ of $\mathbb R^n$, there exists an $n-1$ dimensional affine subspace of $\mathbb R^n$ which bisects each of the $A_i$. In the spirit of the name, this means that given three ham sandwiches of arbitrary size and orientation, I can slice each one in half with a single swipe (supposing, of course, that I have a large enough knife). Alternatively, you could cut a single sandwich such that both halves have half the top bun, half the ham, and half the bottom bun.

What's cool about these theorems is that they seem unrelated, but are really rather immediate results of the Brouwer Fixed Point Theorem. I think Munkres does a good job of covering all of these.

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  • $\begingroup$ These are some nice examples. I think that it's ok that they are all consequences of Brouwer's Fixed Point Theorem because the students won't be able to identify the common root (at least on the first day). $\endgroup$ Aug 12, 2015 at 19:11
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I recently came across Prof. Tokieda's series of lectures on Topology at the African Institute for Mathematical Sciences and found the lectures to be very good as well as very entertaining. The first lecture covers a number of nice examples using Möbius strips.

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  • $\begingroup$ This illustrates gluing of fundamental domains well. $\endgroup$ Aug 12, 2015 at 18:37
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Perhaps not the most elementary, but in Hatcher's Algebraic Topology text (freely available) in the first chapter on the Fundamental Group there is an explanation of two linked circles which lets you intuitively and visually explain a fundamental group which is isomorphic to the additive group of integers (infinite cyclic).

Though you might not even get that far in the class, it shows how something in topology can just be a "geometric translation" of an algebraic concept. It was something that I had read about before I studied topology which was really interesting. It is often helpful to students to show them not what they are going to learn, but what they could learn.

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