Countable choice and pseudometric spaces

In the realm of pseudometric spaces the role of choice principles is investigated. In particular it is shown that in ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the axiom of countable choice is not only sufficient but also necessary to establish each of the following results: 1. separable <----> countable base, 2. separable <----> Lindelof. 3. separable <----> topologically totally bounded, 4. compact --> separable, 5. separability is hereditary, 6. the Baire Category Theorem for complete spaces with countable base, 7. the Baire Category Theorem for complete, totally bounded spaces, 8. compact <----> sequentially compact, 9. compact <----> (totally bounded and complete), 10. sequentially compact <----> (totally bounded and complete), 11. Weierstrass compact <----> (totally bounded and complete). (C) 1998 Elsevier Science B.V.