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

页码：153 / 164

页数：12