Countable choice and pseudometric spaces

被引：17
|
作者：
Bentley, HL
Herrlich, H
机构：
[1] Univ Toledo, Dept Math, Toledo, OH 43606 USA
[2] Univ Bremen, Bremen, Germany
来源：
TOPOLOGY AND ITS APPLICATIONS | 1998年 / 85卷 / 1-3期
关键词：
axiom of (countable) choice; pseudometric space; separable; totally bounded; complete; compact; Baire category;
D O I：
10.1016/S0166-8641(97)00138-7
中图分类号：
O29 [应用数学];
学科分类号：
070104 ;
摘要：
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.
引用
收藏
页码：153 / 164
页数：12
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