Tychonoff’s Theorem

The main goal of these notes is to provide an elementary proof of Tychonoff’s theorem. Tychonoff’s Theorem. The product of arbitrarily many compact spaces is compact. It is easier to prove that the product of finitely many compact spaces is compact than it is to prove the general case. As May says in his notes [6] “The case of finite products is not difficult, but the general case is.” Schaum’s Outline [5] states Tychonoff’s theorem in Chapter 12, but the proof is banished to the exercises. In Munkres’ Topology [7], compactness is introduced in Chapter 3, where it is proved that the product of finitely many compact spaces is compact (Theorem 26.7), and the proof is of the general case for arbitrary products (Theorem 37.3) is postponed until Chapter 5, with a full chapter on countability and separation interrupting. One must use the axiom of choice (or its equivalent) to prove the general case, and it’s perfectly accurate to say