We study conditions for a topological space to be metrizable, properties of metrizable spaces, and the role the axiom of choice plays in these matters.

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.

We study the relationships between definitions of compactness in topological spaces and the roll the axiom of choice plays in these relationships.

We investigate, within the framework of Zermelo-Fraenkel set theory ZF, the interrelations between weak forms of the Axiom of Choice AC restricted to sets of reals.