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. arable ↔ countable base, arable ↔ Lindelöf, arable ↔ topologically totally bounded, pact → separable, arability is hereditary, Baire Category Theorem for complete spaces with countable base, Baire Category Theorem for complete, totally bounded spaces, pact ↔ sequentially compact, pact ↔ (totally bounded and complete), quentially compact ↔ (totally bounded and complete), ierstraβ compact ↔ (totally bounded and complete).