1521-3870/asset/2256_left.gif?v=1&s=becebab9c3e85769a14b145f17b8e337848bc48b)
1521-3870/asset/2256_right.gif?v=1&s=6d251fa351b8e0943ed3dc42b2beb251d4cbfa3e)
Original Paper
Consequences of the failure of the axiom of choice in the theory of Lindelöf metric spaces
Article first published online: 3 FEB 2004
DOI: 10.1002/malq.200310084
Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1521-3870/asset/cover.gif?v=1&s=3df32dbed8d9153ac6eafe1eaa786854b9b48345 "Mathematical Logic Quarterly")
Additional Information
How to Cite
Keremedis, K. (2004), Consequences of the failure of the axiom of choice in the theory of Lindelöf metric spaces. Mathematical Logic Quarterly, 50: 141–151. doi: 10.1002/malq.200310084
Publication History
- Issue published online: 3 FEB 2004
- Article first published online: 3 FEB 2004
- Manuscript Accepted: 20 OCT 2003
- Manuscript Revised: 18 OCT 2003
- Manuscript Received: 6 JUL 2003
- Abstract
- References
- Cited By
Keywords:
- Axiom of choice;
- weak axioms of choice;
- metric space;
- Lindelöf metric space;
- Loeb metric space
Abstract
We study within the framework of Zermelo-Fraenkel set theory ZF the role that the axiom of choice plays in the theory of Lindelöf metric spaces. We show that in ZF the weak choice principles: (i) Every Lindelöf metric space is separable and (ii) Every Lindelöf metric space is second countable (Forms 340 and 341, respectively, in [10]) are equivalent. We also prove that a Lindelöf metric space is hereditarily separable iff it is hereditarily Lindelöf iff it hold as well the axiom of choice restricted to countable sets and to topologies of Lindelöf metric spaces as the countable union theorem restricted to Lindelöf metric spaces. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)