Abstract

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. 1. separable ↔ countable base, 2. 2. separable ↔ Lindelöf, 3. 3. separable ↔ topologically totally bounded, 4. 4. compact → separable, 5. 5. separability is hereditary, 6. 6. the Baire Category Theorem for complete spaces with countable base, 7. 7. the Baire Category Theorem for complete, totally bounded spaces, 8. 8. compact ↔ sequentially compact, 9. 9. compact ↔ (totally bounded and complete), 10. 10. sequentially compact ↔ (totally bounded and complete), 11. 11. Weierstraβ compact ↔ (totally bounded and complete).

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