Just for Fun

Extra Problems: Here are some extra problems that may prove to be more challenging than the homework problems. I do not know the answers to all of them. (If you wish to discuss them, feel free to see me after class or during office hours.)

Extra Problems

Zorn vs Tychonoff: Recall that we used Zorn's Lemma to prove Tychonoff's Theorem, and we mentioned that Zorn's Lemma is equivalent to the Axiom of Choice. It turns out you can go backwards as well: you can use Tychonoff's Theorem to prove the Axiom of Choice. A nice summary of all of these things can be found here (though I did not check this carefully): Notes by John Terilla. Kelley originally published this result in

J. L. Kelley, The Tychonoff product theorem implies the axiom of choice, Fund. Math., pp 75-78, 1950.

Covering Spaces: A delightful summary of covering spaces can be found on the webpage of Professor Dror Bar-Natan; in particular, here.

A Convenient Category of Topological Spaces: We have seen in class that the category of topological spaces admits subspaces, quotients, products, coproducts (disjoint unions), but in general does not admit an exponential law. To obtain this, we must restrict our attention to locally compact Hausdorff spaces, for example. In applications, it is useful to have one category with as many of these properties as possible. The following paper by Steenrod explores a subcategory of topological spaces that has all of these objects and properties.

N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J., 14 (1967), 133--152.