mod07lec43 - Proof of Tychonoff's theorem - Part 1

We finally give a detailed proof of Tychonoff's theorem, using Zorn's Lemma in a fundamental way. As a Corollary, one obtains the fact that (1) a finite Cartesian product of compact spaces is compact, and (2) any closed and bounded subset of R^n is compact, since it is a closed subset of a product of compact intervals in R (which is then compact by (1). We note that both these points can be proved without the full force of Tychonoff's theorem, but here we give them as a corollary.