We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.

We study the Tychonoff Compactness Theorem for several different definitions of a compact space.

This is a continuation of [2]. We study the Tychonoff Compactness Theorem for various definitions of compactness and for various types of spaces . We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.