Category Archives: Teaching

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.”

 

Traces and commutators

The following is a true (and well-known) theorem: \(\newcommand{\Tr}{\mathop{\rm Tr}}\)

Suppose \(A\) and \(B\) are bounded operators on a Hilbert space, and \(AB\) and \(BA\) are trace class.  Then \( \Tr(AB)=\Tr(BA) \).

This is easy to prove if one of the operators \(A,B\) is itself of trace class, or if they both are Hilbert-Schmidt (the obvious calculation works).  In the general case it is a bit harder.  The “usual” argument proceeds via Lidskii’s trace theorem – the trace of any trace-class operator is the sum of the eigenvalues – together with the purely algebraic fact that the nonzero eigenvalues of \(AB\) and \(BA\) are the same (including multiplicities).  Continue reading