Category Archives: Teaching

Topology, Moore or Less – Concluded

We’ve now finished the Moore method topology class that I wrote about in Topology: Moore or Less It’s been an intense experience for everyone, I think.  Many students have surprised themselves by what they have achieved.  At the end of the course we printed off copies of the co-written course textbook (complete with frontispiece photo of the authors!) and everyone received one in time for the “open book” final exam.  I hope that many students will hold on to these as a reminder of our common achievement.  Here’s the cover.

covThe first part of the course was descended from Bing’s notes  (and thus, indirectly, from Moore himself) as they are reproduced in the Journal of Inquiry-Based Learning in Mathematics.  After mid-semester we digressed into product and quotient topologies and then, briefly, into function spaces and the compact-open topology – the last proof that the students were guided through was a special case of the exponential law for function spaces,

\[ X^{Y\times Z} \cong (X^Y)^Z. \]

That seemed like a good place to stop.

Students prepared their in-class presentations in “teams” of 4 rather than individually – this was a modification that I made to the standard Moore method. The online platform Piazza was the main mode of team collaboration – I held a couple of in-class sessions too where teams worked together and I acted as a roving consultant, but I should probably have done more of that.

What did the students think?  Here are a few comments:

“One of the biggest takeaways from this class was seeing how mathematics is constructed firsthand. This semester we constructed a complicated and powerful machine that I am eager to build upon in my later mathematics courses, and now I have the tools necessary to do so.”

“Having to prepare for class with what theorem or example we had to prove for the class and writing our own book has given me an understanding of the material that I don’t think would have happened if the course was taught similar to a traditional course.”

“I felt very engaged in the course as a result of the unique “Moore Method” used to teach the course.”

“The class structure challenged me to think differently than I ever had, and I genuinely appreciated that.”

I love these quotes, but of course not everyone feels the same way. I paraphrase the next comment:

“The method used to teach this class, while I can see its benefits, was really not right for me… I wanted my struggles with the material to be private, and because the class was so collaborative there wasn’t an easy way to do this.”

I hear what this student is saying, and wish I could have helped him/her better. One thing I would have liked to be able to share is that we all struggle, in math as in life, and that I had wanted the class to be a place where we could struggle together, not feel we have to project a brittle confidence or else stay silent.  That will be something to work on for next time, if I do this again.

Thanks, students! I really enjoyed the class and I hope you did too.

Tychonoff’s theorem II

In my earlier post on Tychonoff’s theorem, I talked about the original proof, based on the following characterization of compactness which is due to Kuratowski.

Definition  Let \(S\) be a subset of a topological space \(X\).  A point \(x\in X\) is a point of perfection of \(S\) if, for every neighborhood \(U\) of \(x\), the set \(U\cap S\) has the same cardinality as \(S\).

Lemma (Kuratowski) A topological space \(X\) is compact if and only if every infinite subset has a point of perfection.

Now I will give the proof of this characterization (again following Wright).

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Tychonoff’s theorem

Tychonoff’s theorem (an arbitrary product of compact sets is compact) is one of the high points of any general topology course.  When I’ve taught this in recent years, I’ve usually given the proof using universal nets, which I think is due to Kelley.

Recently though I read a very nice paper by Wright  which reproduces, and then modifies, Tychonoff’s original proof (otherwise inaccessible to me because of my lack of German).  I thought the original proof was really elegant and though I would try to give an exposition.

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Topology, Moore or Less

I get to teach Math 429 this semester.

This is the introductory topology course for undergraduates at Penn State – “point set topology” as the old-fashioned name would be.  I used to teach some of this material at Oxford but I have not had a chance to teach it at PSU before now.  I have about 25 students.

I decided to try a variation of a Moore method approach in this class.

So I started by showing the students the two-minute video above, which shows Steph Davis free-soloing and then BASE jumping from a Utah desert tower. Then I asked them, “Now you have watched the video, could you do that?”

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“Winding Around” is now available

JohnWAI was excited to receive a package from the American Mathematical Society today!  My author’s copies of “Winding Around” have arrived!

This is a book based on by MASS 2013 course of the same title, which looks at the winding number – the central notion in plane topology – from a variety of perspectives, topological, geometrical, analytic and combinatorial.  Having (I hope) made the case that the winding number concept is the “golden cord which guides the student through the labyrinth of classical mathematics”, I conclude by following a beautiful paper of Michael Atiyah to explain how, by asking one natural question about the winding number, we can be led to the Bott periodicity theorem, a central result in the flowering of topology in the 1960s.

If you are interested in the book, please visit the AMS bookstore page.

I will attempt to maintain a list of typos and other corrections here.  At present this list is empty, but I doubt if that happy state of affairs will last for long!  (edit: it didn’t) Please contact me with any information about corrections.  And, to quote the final sentence of the book, “I wish you much happy winding around in the future.”