515 – Homework 2

This set is due Feb. 29 at the beginning of lecture. Let me know if more time is needed or anything like that. Problem 4 was incorrect as stated; I have fixed it now. Thanks to Tara Sheehan for bringing the problem to my attention.

187 – List of all presentations

For ease, I re-list here all the presentations we had throughout the term. I also include some of them. If you gave a presentation and would like your notes to be included, please email them to me and I’ll add them here.

• Jeremy Elison, Wednesday, October 12: Georg Cantor and infinity.
• Kevin Byrne, Wednesday, October 26: Alan Turing and Turing machines.
• Keith Ward, Monday, November 7: Grigori Perelman and the Poincaré conjecture.
• David Miller, Wednesday, November 16: Augustin Cauchy and Cauchy’s dispersion equation.
• Taylor Mitchell, Friday, November 18: Lajos Pósa and Hamiltonian circuits.
• Sheryl Tremble, Monday, November 28: Pythagoras and the Pythagorean theorem.
• Blake Dietz, Wednesday, November 30: and the Happy End problem.

Here are Jeremy’s notes on his presentation. Here is the Wikipedia page on Cantor, and a link to Cantor’s Attic, a wiki-style page discussing the different (set theoretic) notions of infinity.

Here are a link to the official page for the Alan Turing year, and the Wikipedia page on Turing. If you have heard of Conway’s Game of Life, you may enjoy the following video showing how to simulate a Turing machine within the Game of Life; the Droste effect it refers to is best explained in by H. Lenstra in a talk given at Princeton on April 3, 2007, and available here.

Here is a link to the Wikipedia page on Perelman, and the Clay Institute’s description of the Poincaré conjecture. In 2006, The New Yorker published an interesting article on the unfortunate “controversy” on the priority of Perelman’s proof.

Here are David’s slides on his presentation, and the Wikipedia page on Cauchy.

Here is a link to Ross Honsberger’s article on Pósa (including the result on Hamiltonian circuits that Taylor showed during her presentation).

Here are Sheryl’s slides on Pythagoras and his theorem. In case the gif file does not play, here is a separate copy:

The Pythagorean theorem has many proofs, even one discovered by President Garfield!

Finally, here is the Wikipedia page on . Oakland University has a nice page on him, including information on the number; see also the page maintained by Peter Komjáth, and an online depository of most of papers.

502 – Equivalents of the axiom of choice

The goal of this note is to show the following result:

Theorem 1 The following statements are equivalent in

 

1. The axiom of choice: Every set can be well-ordered.
2. Every collection of nonempty set admits a choice function, i.e., if for all then there is such that for all
3. Zorn’s lemma: If is a partially ordered set with the property that every chain has an upper bound, then has maximal elements.
4. Any family of pairwise disjoint nonempty sets admits a selector, i.e., a set such that for all in the family.
5. Any set is a well-ordered union of finite sets of bounded size, i.e., for every set there is a natural an ordinal and a function such that for all and
6. Tychonoff’s theorem: The topological product of compact spaces is compact.
7. Every vector space (over any field) admits a basis.

502 – Cantor-Bendixson derivatives

Given a topological space and a set let be the set of accumulation points of i.e., those points of such that any open neighborhood of meets in an infinite set.

Suppose that is closed. Then Define for closed compact by recursion: and for limit. Note that this is a decreasing sequence, so that if we set there must be an such that for all  

[The sets are the Cantor-Bendixson derivatives of In general, a derivative operation is a way of associating to sets some kind of “boundary.”]